Господин Экзамен

Lang:

Другие калькуляторы

Как пользоваться?
c^2-6 если c=1/4
cos(sin(x)) если x=-1/4
log(2)*2*x+12*log(2)*x+20 если x=-1
sin(x)^2-tan(x)*cot(x) если x=1/3
Разложить многочлен на множители:
20*a^2-68*a+45
16*a^2-9*b^2*c^2
2*x^2+x-21
x^2-12*x-36
Общий знаменатель:
(y^2-10)/(y-8)-(54)/(y-8)
25/a^2-5*a+25+2*a/5+a+a^3-25*a^2/a^3+125
(1/(1-tan(x)))-(1/(1+tan(x)))
(c+4)/(a-3)/(c^2+8*c+64)/(a^2-9)
Умножение столбиком:
46*17
54*13
95*95
845*100
Деление столбиком:
58/4
196/5
78500/5
87993/5
Раскрыть скобки в:
(a-4*c+6*d-n)*(-4)
a^3*(a^2-7)^2-36*a
4*k^(4*k^2+2*k-4)+8*k^(2*k^2+k-2)
8*c*(p-12)+7*d*(p-12)^2
Разложение числа на простые множители:
2816
3960
770
8450
Идентичные выражения
cos(sin(x)) если x=- один / четыре
косинус от ( синус от (x)) если x равно минус 1 делить на 4
косинус от ( синус от (x)) если x равно минус один делить на четыре
cossinx если x=-1/4
cos(sin(x)) при x=-1/4
cos(sin(x)) если x=-1 разделить на 4
Похожие выражения
cos(sin(x)) если x=+1/4
cos(sinx) если x=-1/4

Упрощение выражений/ cos(sin(x)) если x=-1/4

cos(sin(x)) если x=-1/4

Решение

Вы ввели [src]

cos(sin(x))

$$\cos{\left(\sin{\left(x \right)} \right)}$$

cos(sin(x))

Подстановка условия [src]

cos(sin(x)) при x = -1/4

подставляем

cos(sin(x))

$$\cos{\left(\sin{\left(x \right)} \right)}$$

cos(sin(x))

$$\cos{\left(\sin{\left(x \right)} \right)}$$

переменные

x = -1/4

$$x = - \frac{1}{4}$$

cos(sin((-1/4)))

$$\cos{\left(\sin{\left((-1/4) \right)} \right)}$$

cos(sin(-1/4))

$$\cos{\left(\sin{\left(- \frac{1}{4} \right)} \right)}$$

cos(sin(1/4))

$$\cos{\left(\sin{\left(\frac{1}{4} \right)} \right)}$$

cos(sin(1/4))

Степени [src]

  I*x    -I*x     -I*x    I*x
 e      e        e       e   
 ---- - -----    ----- - ----
  2       2        2      2  
e               e            
------------- + -------------
      2               2

$$\frac{e^{- \frac{e^{i x}}{2} + \frac{e^{- i x}}{2}}}{2} + \frac{e^{\frac{e^{i x}}{2} - \frac{e^{- i x}}{2}}}{2}$$

exp(exp(i*x)/2 - exp(-i*x)/2)/2 + exp(exp(-i*x)/2 - exp(i*x)/2)/2

Численный ответ [src]

cos(sin(x))

cos(sin(x))

Тригонометрическая часть [src]

   /   /    pi\\
cos|cos|x - --||
   \   \    2 //

$$\cos{\left(\cos{\left(x - \frac{\pi}{2} \right)} \right)}$$

   /pi         \
sin|-- + sin(x)|
   \2          /

$$\sin{\left(\sin{\left(x \right)} + \frac{\pi}{2} \right)}$$

     1     
-----------
   /  1   \
sec|------|
   \csc(x)/

$$\frac{1}{\sec{\left(\frac{1}{\csc{\left(x \right)}} \right)}}$$

       1        
----------------
   /pi     1   \
csc|-- - ------|
   \2    csc(x)/

$$\frac{1}{\csc{\left(\frac{\pi}{2} - \frac{1}{\csc{\left(x \right)}} \right)}}$$

       1        
----------------
   /     1     \
sec|-----------|
   |   /    pi\|
   |sec|x - --||
   \   \    2 //

$$\frac{1}{\sec{\left(\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}} \right)}}$$

       1        
----------------
   /     1     \
sec|-----------|
   |   /pi    \|
   |sec|-- - x||
   \   \2     //

$$\frac{1}{\sec{\left(\frac{1}{\sec{\left(- x + \frac{\pi}{2} \right)}} \right)}}$$

          1          
---------------------
   /pi        1     \
csc|-- - -----------|
   \2    csc(pi - x)/

$$\frac{1}{\csc{\left(\frac{\pi}{2} - \frac{1}{\csc{\left(- x + \pi \right)}} \right)}}$$

   /                /x\\
cos|(1 + cos(x))*tan|-||
   \                \2//

$$\cos{\left(\left(\cos{\left(x \right)} + 1\right) \tan{\left(\frac{x}{2} \right)} \right)}$$

                     /pi + 2*sin(x)\
(1 - sin(sin(x)))*tan|-------------|
                     \      4      /

$$\left(- \sin{\left(\sin{\left(x \right)} \right)} + 1\right) \tan{\left(\frac{2 \sin{\left(x \right)} + \pi}{4} \right)}$$

                     /pi - 2*sin(x)\
(1 - sin(sin(x)))*cot|-------------|
                     \      4      /

$$\left(- \sin{\left(\sin{\left(x \right)} \right)} + 1\right) \cot{\left(\frac{- 2 \sin{\left(x \right)} + \pi}{4} \right)}$$

           /          1                         sin(x)    \
           |2 - -------------- + 2*sin(x) - --------------|
           |       2/pi + 2*x\                 2/pi + 2*x\|
           |    sin |--------|              sin |--------||
          2|        \   4    /                  \   4    /|
-1 + 2*cos |----------------------------------------------|
           \                      4                       /

$$2 \cos^{2}{\left(\frac{2 \sin{\left(x \right)} + 2 - \frac{\sin{\left(x \right)}}{\sin^{2}{\left(\frac{2 x + \pi}{4} \right)}} - \frac{1}{\sin^{2}{\left(\frac{2 x + \pi}{4} \right)}}}{4} \right)} - 1$$

        /      /x\  \
        |   tan|-|  |
       2|      \2/  |
1 - tan |-----------|
        |       2/x\|
        |1 + tan |-||
        \        \2//
---------------------
        /      /x\  \
        |   tan|-|  |
       2|      \2/  |
1 + tan |-----------|
        |       2/x\|
        |1 + tan |-||
        \        \2//

$$\frac{- \tan^{2}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} \right)} + 1}{\tan^{2}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} \right)} + 1}$$

      /           /x\  \  
      |        cot|-|  |  
      |pi         \2/  |  
 2*cot|-- - -----------|  
      |4           2/x\|  
      |     1 + cot |-||  
      \             \2//  
--------------------------
        /           /x\  \
        |        cot|-|  |
       2|pi         \2/  |
1 + cot |-- - -----------|
        |4           2/x\|
        |     1 + cot |-||
        \             \2//

$$\frac{2 \cot{\left(\frac{\pi}{4} - \frac{\cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} \right)}}{\cot^{2}{\left(\frac{\pi}{4} - \frac{\cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} \right)} + 1}$$

      /           /x\  \  
      |        tan|-|  |  
      |pi         \2/  |  
 2*tan|-- + -----------|  
      |4           2/x\|  
      |     1 + tan |-||  
      \             \2//  
--------------------------
        /           /x\  \
        |        tan|-|  |
       2|pi         \2/  |
1 + tan |-- + -----------|
        |4           2/x\|
        |     1 + tan |-||
        \             \2//

$$\frac{2 \tan{\left(\frac{\pi}{4} + \frac{\tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} \right)}}{\tan^{2}{\left(\frac{\pi}{4} + \frac{\tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} \right)} + 1}$$

     /                /x\\                
  cos|(1 + cos(x))*tan|-|| + cos(sin(x))  
     \                \2//                
------------------------------------------
                     /                /x\\
2 - cos(sin(x)) + cos|(1 + cos(x))*tan|-||
                     \                \2//

$$\frac{\cos{\left(\left(\cos{\left(x \right)} + 1\right) \tan{\left(\frac{x}{2} \right)} \right)} + \cos{\left(\sin{\left(x \right)} \right)}}{\cos{\left(\left(\cos{\left(x \right)} + 1\right) \tan{\left(\frac{x}{2} \right)} \right)} - \cos{\left(\sin{\left(x \right)} \right)} + 2}$$

                1             
1 - --------------------------
       2/         1          \
    cot |--------------------|
        |/       1   \    /x\|
        ||1 + -------|*cot|-||
        ||       2/x\|    \2/|
        ||    cot |-||       |
        \\        \2//       /
------------------------------
                1             
1 + --------------------------
       2/         1          \
    cot |--------------------|
        |/       1   \    /x\|
        ||1 + -------|*cot|-||
        ||       2/x\|    \2/|
        ||    cot |-||       |
        \\        \2//       /

$$\frac{1 - \frac{1}{\cot^{2}{\left(\frac{1}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right) \cot{\left(\frac{x}{2} \right)}} \right)}}}{1 + \frac{1}{\cot^{2}{\left(\frac{1}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right) \cot{\left(\frac{x}{2} \right)}} \right)}}}$$

         /         2/x   pi\  \
         | -1 + tan |- + --|  |
        2|          \2   4 /  |
-1 + cot |--------------------|
         |  /       2/x   pi\\|
         |2*|1 + tan |- + --|||
         \  \        \2   4 ///
-------------------------------
         /         2/x   pi\  \
         | -1 + tan |- + --|  |
        2|          \2   4 /  |
 1 + cot |--------------------|
         |  /       2/x   pi\\|
         |2*|1 + tan |- + --|||
         \  \        \2   4 ///

$$\frac{\cot^{2}{\left(\frac{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1}{2 \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)} \right)} - 1}{\cot^{2}{\left(\frac{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1}{2 \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)} \right)} + 1}$$

        /         2/x   pi\  \
        |  1 - cot |- + --|  |
       2|          \2   4 /  |
1 - tan |--------------------|
        |  /       2/x   pi\\|
        |2*|1 + cot |- + --|||
        \  \        \2   4 ///
------------------------------
        /         2/x   pi\  \
        |  1 - cot |- + --|  |
       2|          \2   4 /  |
1 + tan |--------------------|
        |  /       2/x   pi\\|
        |2*|1 + cot |- + --|||
        \  \        \2   4 ///

$$\frac{- \tan^{2}{\left(\frac{- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1}{2 \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)} \right)} + 1}{\tan^{2}{\left(\frac{- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1}{2 \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)} \right)} + 1}$$

/                                                                            /      True         for x mod pi = 0
|                                                                            |                                   
|                                  1                                     for

$$\begin{cases} 1 & \text{for}\: \operatorname{ITE}\left(x \bmod \pi = 0, \text{True}, \frac{\sin{\left(x \right)}}{2} \bmod \pi = 0\right) \\\left(\left(\cot^{2}{\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}}{2} \right)}\right) - 1\right) \left(\sin^{2}{\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}}{2} \right)}\right) & \text{otherwise} \end{cases}$$

/                                                                           /      True         for x mod pi = 0
|                                                                           |                                   
|                                  1                                    for

$$\begin{cases} 1 & \text{for}\: \operatorname{ITE}\left(x \bmod \pi = 0, \text{True}, \frac{\sin{\left(x \right)}}{2} \bmod \pi = 0\right) \\\left(\left(- \tan^{2}{\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}}{2} \right)}\right) + 1\right) \left(\cos^{2}{\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}}{2} \right)}\right) & \text{otherwise} \end{cases}$$

/                                         /         False            for x mod pi = 0
|                                         |                                          
|                 0                   for

$$\begin{cases} 0 & \text{for}\: \operatorname{ITE}\left(x \bmod \pi = 0, \text{False}, \left(\sin{\left(x \right)} + \frac{\pi}{2}\right) \bmod \pi = 0\right) \\\frac{\left(\sin{\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases} \right)}\right) + 1}{\tan{\left(\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}}{2}\right) + \frac{\pi}{4} \right)}} & \text{otherwise} \end{cases}$$

                                          /          /       2/x\        \\
                                          |          |  2*sin |-|*sin(x) ||
                                          |         4|        \2/        ||
                                          |    4*sin |-------------------||
                                          |          |   2           4/x\||
                                        2 |          |sin (x) + 4*sin |-|||
/                     2                \  |          \                \2//|
|-1 + ---------------------------------| *|1 - ---------------------------|
|        2/           sin(x)          \|  |         /       2/x\        \ |
|     sin |---------------------------||  |         |  4*sin |-|*sin(x) | |
|         |/   1         4   \    2/x\||  |        2|        \2/        | |
|         ||------- + -------|*sin |-|||  |     sin |-------------------| |
|         ||   2         4/x\|     \2/||  |         |   2           4/x\| |
|         ||sin (x)   sin |-||        ||  |         |sin (x) + 4*sin |-|| |
\         \\              \2//        //  \         \                \2// /

$$\left(-1 + \frac{2}{\sin^{2}{\left(\frac{\sin{\left(x \right)}}{\left(\frac{1}{\sin^{2}{\left(x \right)}} + \frac{4}{\sin^{4}{\left(\frac{x}{2} \right)}}\right) \sin^{2}{\left(\frac{x}{2} \right)}} \right)}}\right)^{2} \left(- \frac{4 \sin^{4}{\left(\frac{2 \sin^{2}{\left(\frac{x}{2} \right)} \sin{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)} + \sin^{2}{\left(x \right)}} \right)}}{\sin^{2}{\left(\frac{4 \sin^{2}{\left(\frac{x}{2} \right)} \sin{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)} + \sin^{2}{\left(x \right)}} \right)}} + 1\right)$$

/                                                                                                  /                         /    3*pi\             
|                                                                                                  |       False         for |x + ----| mod 2*pi = 0
|                                             1                                                for <                         \     2  /             
|                                                                                                  |                                                
|                                                                                                  \sin(x) mod 2*pi = 0           otherwise         
|                                                                                                                                                   
<    //            /    3*pi\             \ /         //            /    3*pi\             \\                                                       
|    ||  1     for |x + ----| mod 2*pi = 0| |         ||  1     for |x + ----| mod 2*pi = 0||                                                       
|    |<            \     2  /             | |         |<            \     2  /             ||                                                       
|    ||                                   | |         ||                                   ||                                                       
|   2|\sin(x)           otherwise         | |        2|\sin(x)           otherwise         ||                                                       
|sin |------------------------------------|*|-1 + cot |------------------------------------||                        otherwise                      
\    \                 2                  / \         \                 2                  //

$$\begin{cases} 1 & \text{for}\: \operatorname{ITE}\left(\left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0, \text{False}, \sin{\left(x \right)} \bmod 2 \pi = 0\right) \\\left(\left(\cot^{2}{\left(\frac{\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}}{2} \right)}\right) - 1\right) \left(\sin^{2}{\left(\frac{\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}}{2} \right)}\right) & \text{otherwise} \end{cases}$$

/                                                                      /      True         for x mod pi = 0
|                                                                      |                                   
|                               1                                  for

$$\begin{cases} 1 & \text{for}\: \operatorname{ITE}\left(x \bmod \pi = 0, \text{True}, \frac{\sin{\left(x \right)}}{2} \bmod \pi = 0\right) \\\frac{2 \left(\sin^{2}{\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}}{2} \right)}\right) \left(\cos{\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases} \right)}\right)}{\left(- \cos{\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases} \right)}\right) + 1} & \text{otherwise} \end{cases}$$

/                                                                                   /        True           for x mod pi = 0
|                                                                                   |                                       
|                                                                                   |1 - cos(x)                             
|                                      1                                        for <---------- mod pi = 0     otherwise    
|                                                                                   |      /x\                              
|                                                                                   | 2*tan|-|                              
|                                                                                   \      \2/                              
|                                                                                                                           
<    //    0       for x mod pi = 0\ /        //    0       for x mod pi = 0\\                                              
|    ||                            | |        ||                            ||                                              
|    ||1 - cos(x)                  | |        ||1 - cos(x)                  ||                                              
|    |<----------     otherwise    | |        |<----------     otherwise    ||                                              
|    ||     /x\                    | |        ||     /x\                    ||                                              
|    ||  tan|-|                    | |        ||  tan|-|                    ||                                              
|   2|\     \2/                    | |       2|\     \2/                    ||                                              
|cos |-----------------------------|*|1 - tan |-----------------------------||                   otherwise                  
\    \              2              / \        \              2              //

$$\begin{cases} 1 & \text{for}\: \operatorname{ITE}\left(x \bmod \pi = 0, \text{True}, \frac{- \cos{\left(x \right)} + 1}{2 \tan{\left(\frac{x}{2} \right)}} \bmod \pi = 0\right) \\\left(\left(- \tan^{2}{\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{- \cos{\left(x \right)} + 1}{\tan{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}}{2} \right)}\right) + 1\right) \left(\cos^{2}{\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{- \cos{\left(x \right)} + 1}{\tan{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}}{2} \right)}\right) & \text{otherwise} \end{cases}$$

/                                                                              /      True         for x mod pi = 0
|                                                                              |                                   
|                                   1                                      for

$$\begin{cases} 1 & \text{for}\: \operatorname{ITE}\left(x \bmod \pi = 0, \text{True}, \frac{\sin{\left(x \right)}}{2} \bmod \pi = 0\right) \\\left(-1 + \left(\frac{\sin^{2}{\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases} \right)}}{4 \left(\sin^{4}{\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}}{2} \right)}\right)}\right)\right) \left(\sin^{2}{\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}}{2} \right)}\right) & \text{otherwise} \end{cases}$$

/                                                                                                              /      True         for x mod pi = 0
|                                                                                                              |                                   
|                                                   1                                                      for

$$\begin{cases} 1 & \text{for}\: \operatorname{ITE}\left(x \bmod \pi = 0, \text{True}, \frac{\sin{\left(x \right)}}{2} \bmod \pi = 0\right) \\4 \left(-1 + \left(\frac{1}{\tan^{2}{\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}}{2} \right)}}\right)\right) \left(\cos^{4}{\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}}{4} \right)}\right) \left(\tan^{2}{\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}}{4} \right)}\right) & \text{otherwise} \end{cases}$$

/                                               /         True           for x mod pi = 0
|                                               |                                        
|                                               |      /x\                               
|                                               |   cot|-|                               
|                    1                      for <      \2/                               
|                                               |----------- mod pi = 0     otherwise    
|                                               |       2/x\                             
|                                               |1 + cot |-|                             
|                                               \        \2/                             
|                                                                                        
|         //     0       for x mod pi = 0\                                               
|         ||                             |                                               
|         ||       /x\                   |                                               
|         ||  2*cot|-|                   |                                               
|         |<       \2/                   |                                               
|         ||-----------     otherwise    |                                               
<         ||       2/x\                  |                                               
|         ||1 + cot |-|                  |                                               
|        2|\        \2/                  |                                               
|-1 + cot |------------------------------|                                               
|         \              2               /                                               
|-----------------------------------------                    otherwise                  
|         //     0       for x mod pi = 0\                                               
|         ||                             |                                               
|         ||       /x\                   |                                               
|         ||  2*cot|-|                   |                                               
|         |<       \2/                   |                                               
|         ||-----------     otherwise    |                                               
|         ||       2/x\                  |                                               
|         ||1 + cot |-|                  |                                               
|        2|\        \2/                  |                                               
| 1 + cot |------------------------------|                                               
\         \              2               /

$$\begin{cases} 1 & \text{for}\: \operatorname{ITE}\left(x \bmod \pi = 0, \text{True}, \frac{\cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} \bmod \pi = 0\right) \\\frac{\left(\cot^{2}{\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}}{2} \right)}\right) - 1}{\left(\cot^{2}{\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}}{2} \right)}\right) + 1} & \text{otherwise} \end{cases}$$

/                                                                           /        True           for x mod pi = 0
|                                                                           |                                       
|                                                                           |1 - cos(x)                             
|                                  1                                    for <---------- mod pi = 0     otherwise    
|                                                                           |      /x\                              
|                                                                           | 2*tan|-|                              
|                                                                           \      \2/                              
<                                                                                                                   
|  /       //  0     for x mod pi = 0\\    //  0     for x mod pi = 0\                                              
|2*|1 - cos|<                        ||*cos|<                        |                                              
|  \       \\sin(x)     otherwise    //    \\sin(x)     otherwise    /                                              
|---------------------------------------------------------------------                   otherwise                  
|                          //  0     for x mod pi = 0\                                                              
|                 2 - 2*cos|<                        |                                                              
\                          \\sin(x)     otherwise    /

$$\begin{cases} 1 & \text{for}\: \operatorname{ITE}\left(x \bmod \pi = 0, \text{True}, \frac{- \cos{\left(x \right)} + 1}{2 \tan{\left(\frac{x}{2} \right)}} \bmod \pi = 0\right) \\\frac{2 \cdot \left(\left(- \cos{\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases} \right)}\right) + 1\right) \left(\cos{\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases} \right)}\right)}{\left(- 2 \left(\cos{\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases} \right)}\right)\right) + 2} & \text{otherwise} \end{cases}$$

/                                               /       True          for x mod pi = 0
|                                               |                                     
|                    1                      for <   1                                 
|                                               |-------- mod pi = 0     otherwise    
|                                               \2*csc(x)                             
|                                                                                     
|           //  0     for x mod pi = 0\                                               
|           ||                        |                                               
|           |<  1                     |                                               
|           ||------     otherwise    |                                               
|          2|\csc(x)                  |                                               
|       csc |-------------------------|                                               
|           \            2            /                                               
|-1 + ------------------------------------                                            
<         /     /  0     for x mod pi = 0\                                            
|         |     |                        |                                            
|         |     <  1                     |                                            
|         |     |------     otherwise    |                                            
|        2|pi   \csc(x)                  |                                            
|     csc |-- - -------------------------|                                            
|         \2                2            /                                            
|-----------------------------------------                  otherwise                 
|         //  0     for x mod pi = 0\                                                 
|         ||                        |                                                 
|         |<  1                     |                                                 
|         ||------     otherwise    |                                                 
|        2|\csc(x)                  |                                                 
|     csc |-------------------------|                                                 
\         \            2            /

$$\begin{cases} 1 & \text{for}\: \operatorname{ITE}\left(x \bmod \pi = 0, \text{True}, \frac{1}{2 \csc{\left(x \right)}} \bmod \pi = 0\right) \\\frac{-1 + \left(\frac{\csc^{2}{\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\csc{\left(x \right)}} & \text{otherwise} \end{cases}}{2} \right)}}{\csc^{2}{\left(\left(- \frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\csc{\left(x \right)}} & \text{otherwise} \end{cases}}{2}\right) + \frac{\pi}{2} \right)}}\right)}{\csc^{2}{\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\csc{\left(x \right)}} & \text{otherwise} \end{cases}}{2} \right)}} & \text{otherwise} \end{cases}$$

/                                               /       True          for x mod pi = 0
|                                               |                                     
|                    1                      for <   1                                 
|                                               |-------- mod pi = 0     otherwise    
|                                               \2*csc(x)                             
|                                                                                     
|         //  0     for x mod pi = 0     \                                            
|         ||                             |                                            
|         |<  1                          |                                            
|         ||------     otherwise         |                                            
|        2|\csc(x)                     pi|                                            
|     sec |------------------------- - --|                                            
|         \            2               2 /                                            
|-1 + ------------------------------------                                            
|           //  0     for x mod pi = 0\                                               
<           ||                        |                                               
|           |<  1                     |                                               
|           ||------     otherwise    |                                               
|          2|\csc(x)                  |                                               
|       sec |-------------------------|                                               
|           \            2            /                                               
|-----------------------------------------                  otherwise                 
|    //     0       for x mod pi = 0     \                                            
|    ||                                  |                                            
|    ||     1                            |                                            
|    |<-----------     otherwise         |                                            
|    ||   /    pi\                       |                                            
|    ||sec|x - --|                       |                                            
|   2|\   \    2 /                     pi|                                            
|sec |------------------------------ - --|                                            
\    \              2                  2 /

$$\begin{cases} 1 & \text{for}\: \operatorname{ITE}\left(x \bmod \pi = 0, \text{True}, \frac{1}{2 \sec{\left(x - \frac{\pi}{2} \right)}} \bmod \pi = 0\right) \\\frac{\left(\frac{\sec^{2}{\left(\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\csc{\left(x \right)}} & \text{otherwise} \end{cases}}{2}\right) - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\csc{\left(x \right)}} & \text{otherwise} \end{cases}}{2} \right)}}\right) - 1}{\sec^{2}{\left(\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}}{2}\right) - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}$$

/                                                   /            False              for x mod pi = 0
|                                                   |                                               
|                                                   |/            /x\ \                             
|                                                   ||       2*cot|-| |                             
|                      0                        for <|pi          \2/ |                             
|                                                   ||-- + -----------| mod pi = 0     otherwise    
|                                                   ||2           2/x\|                             
|                                                   ||     1 + cot |-||                             
|                                                   \\             \2//                             
|                                                                                                   
|       //     0       for x mod pi = 0     \                                                       
|       ||                                  |                                                       
|       ||       /x\                        |                                                       
|       ||  2*cot|-|                        |                                                       
|       |<       \2/                        |                                                       
|       ||-----------     otherwise         |                                                       
<       ||       2/x\                       |                                                       
|       ||1 + cot |-|                       |                                                       
|       |\        \2/                     pi|                                                       
|  2*cot|------------------------------ + --|                                                       
|       \              2                  4 /                                                       
|---------------------------------------------                       otherwise                      
|        //     0       for x mod pi = 0     \                                                      
|        ||                                  |                                                      
|        ||       /x\                        |                                                      
|        ||  2*cot|-|                        |                                                      
|        |<       \2/                        |                                                      
|        ||-----------     otherwise         |                                                      
|        ||       2/x\                       |                                                      
|        ||1 + cot |-|                       |                                                      
|       2|\        \2/                     pi|                                                      
|1 + cot |------------------------------ + --|                                                      
\        \              2                  4 /

$$\begin{cases} 0 & \text{for}\: \operatorname{ITE}\left(x \bmod \pi = 0, \text{False}, \left(\frac{\pi}{2} + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \bmod \pi = 0\right) \\\frac{2 \left(\cot{\left(\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}}{2}\right) + \frac{\pi}{4} \right)}\right)}{\left(\cot^{2}{\left(\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}}{2}\right) + \frac{\pi}{4} \right)}\right) + 1} & \text{otherwise} \end{cases}$$

/                                                        /             True                for x mod pi = 0
|                                                        |                                                 
|                                                        |         1                                       
|                                                        |-------------------- mod pi = 0     otherwise    
|                        1                           for

$$\begin{cases} 1 & \text{for}\: \operatorname{ITE}\left(x \bmod \pi = 0, \text{True}, \frac{1}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right) \tan{\left(\frac{x}{2} \right)}} \bmod \pi = 0\right) \\\frac{-1 + \left(\frac{1}{\tan^{2}{\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right) \tan{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}}{2} \right)}}\right)}{1 + \left(\frac{1}{\tan^{2}{\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right) \tan{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}}{2} \right)}}\right)} & \text{otherwise} \end{cases}$$

/                                                                                                /      True         for x mod pi = 0
|                                                                                                |                                   
|                                            1                                               for

$$\begin{cases} 1 & \text{for}\: \operatorname{ITE}\left(x \bmod \pi = 0, \text{True}, \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{2} \bmod \pi = 0\right) \\\left(-1 + \left(\frac{\cos^{2}{\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\cos{\left(x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}}{2} \right)}}{\cos^{2}{\left(\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\cos{\left(x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}}{2}\right) - \frac{\pi}{2} \right)}}\right)\right) \left(\cos^{2}{\left(\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\cos{\left(x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}}{2}\right) - \frac{\pi}{2} \right)}\right) & \text{otherwise} \end{cases}$$

          /           2/x\       \
          |      2*sin |-|       |
         4|            \2/       |
    4*sin |----------------------|
          |/         4/x\\       |
          ||    4*sin |-||       |
          ||          \2/|       |
          ||1 + ---------|*sin(x)|
          ||        2    |       |
          \\     sin (x) /       /
1 - ------------------------------
         /           2/x\       \ 
         |      4*sin |-|       | 
        2|            \2/       | 
     sin |----------------------| 
         |/         4/x\\       | 
         ||    4*sin |-||       | 
         ||          \2/|       | 
         ||1 + ---------|*sin(x)| 
         ||        2    |       | 
         \\     sin (x) /       / 
----------------------------------
          /           2/x\       \
          |      2*sin |-|       |
         4|            \2/       |
    4*sin |----------------------|
          |/         4/x\\       |
          ||    4*sin |-||       |
          ||          \2/|       |
          ||1 + ---------|*sin(x)|
          ||        2    |       |
          \\     sin (x) /       /
1 + ------------------------------
         /           2/x\       \ 
         |      4*sin |-|       | 
        2|            \2/       | 
     sin |----------------------| 
         |/         4/x\\       | 
         ||    4*sin |-||       | 
         ||          \2/|       | 
         ||1 + ---------|*sin(x)| 
         ||        2    |       | 
         \\     sin (x) /       /

$$\frac{- \frac{4 \sin^{4}{\left(\frac{2 \sin^{2}{\left(\frac{x}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right) \sin{\left(x \right)}} \right)}}{\sin^{2}{\left(\frac{4 \sin^{2}{\left(\frac{x}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right) \sin{\left(x \right)}} \right)}} + 1}{\frac{4 \sin^{4}{\left(\frac{2 \sin^{2}{\left(\frac{x}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right) \sin{\left(x \right)}} \right)}}{\sin^{2}{\left(\frac{4 \sin^{2}{\left(\frac{x}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right) \sin{\left(x \right)}} \right)}} + 1}$$

            /          /x\       \   
            |       sec|-|       |   
           2|          \2/       |   
        sec |--------------------|   
            |/       2/x\\       |   
            ||    sec |-||       |   
            ||        \2/|    /x\|   
            ||1 + -------|*csc|-||   
            ||       2/x\|    \2/|   
            ||    csc |-||       |   
            \\        \2//       /   
1 - ---------------------------------
        /                 /x\       \
        |              sec|-|       |
       2|  pi             \2/       |
    sec |- -- + --------------------|
        |  2    /       2/x\\       |
        |       |    sec |-||       |
        |       |        \2/|    /x\|
        |       |1 + -------|*csc|-||
        |       |       2/x\|    \2/|
        |       |    csc |-||       |
        \       \        \2//       /
-------------------------------------
            /          /x\       \   
            |       sec|-|       |   
           2|          \2/       |   
        sec |--------------------|   
            |/       2/x\\       |   
            ||    sec |-||       |   
            ||        \2/|    /x\|   
            ||1 + -------|*csc|-||   
            ||       2/x\|    \2/|   
            ||    csc |-||       |   
            \\        \2//       /   
1 + ---------------------------------
        /                 /x\       \
        |              sec|-|       |
       2|  pi             \2/       |
    sec |- -- + --------------------|
        |  2    /       2/x\\       |
        |       |    sec |-||       |
        |       |        \2/|    /x\|
        |       |1 + -------|*csc|-||
        |       |       2/x\|    \2/|
        |       |    csc |-||       |
        \       \        \2//       /

$$\frac{- \frac{\sec^{2}{\left(\frac{\sec{\left(\frac{x}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right) \csc{\left(\frac{x}{2} \right)}} \right)}}{\sec^{2}{\left(- \frac{\pi}{2} + \frac{\sec{\left(\frac{x}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right) \csc{\left(\frac{x}{2} \right)}} \right)}} + 1}{\frac{\sec^{2}{\left(\frac{\sec{\left(\frac{x}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right) \csc{\left(\frac{x}{2} \right)}} \right)}}{\sec^{2}{\left(- \frac{\pi}{2} + \frac{\sec{\left(\frac{x}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right) \csc{\left(\frac{x}{2} \right)}} \right)}} + 1}$$

/                                                                /                                    /    3*pi\             
|                                                                |            False               for |x + ----| mod 2*pi = 0
|                                                                |                                    \     2  /             
|                                                                |                                                           
|                                                                |        2/x   pi\                                          
|                            1                               for <-1 + tan |- + --|                                          
|                                                                |         \2   4 /                                          
|                                                                |----------------- mod 2*pi = 0           otherwise         
|                                                                |        2/x   pi\                                          
|                                                                | 1 + tan |- + --|                                          
|                                                                \         \2   4 /                                          
|                                                                                                                            
|         //                       /    3*pi\             \                                                                  
|         ||        1          for |x + ----| mod 2*pi = 0|                                                                  
|         ||                       \     2  /             |                                                                  
|         ||                                              |                                                                  
|         ||        2/x   pi\                             |                                                                  
|         |<-1 + tan |- + --|                             |                                                                  
|         ||         \2   4 /                             |                                                                  
<         ||-----------------           otherwise         |                                                                  
|         ||        2/x   pi\                             |                                                                  
|         || 1 + tan |- + --|                             |                                                                  
|        2|\         \2   4 /                             |                                                                  
|-1 + cot |-----------------------------------------------|                                                                  
|         \                       2                       /                                                                  
|----------------------------------------------------------                             otherwise                            
|        //                       /    3*pi\             \                                                                   
|        ||        1          for |x + ----| mod 2*pi = 0|                                                                   
|        ||                       \     2  /             |                                                                   
|        ||                                              |                                                                   
|        ||        2/x   pi\                             |                                                                   
|        |<-1 + tan |- + --|                             |                                                                   
|        ||         \2   4 /                             |                                                                   
|        ||-----------------           otherwise         |                                                                   
|        ||        2/x   pi\                             |                                                                   
|        || 1 + tan |- + --|                             |                                                                   
|       2|\         \2   4 /                             |                                                                   
|1 + cot |-----------------------------------------------|                                                                   
\        \                       2                       /

$$\begin{cases} 1 & \text{for}\: \operatorname{ITE}\left(\left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0, \text{False}, \frac{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1} \bmod 2 \pi = 0\right) \\\frac{\left(\cot^{2}{\left(\frac{\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}}{2} \right)}\right) - 1}{\left(\cot^{2}{\left(\frac{\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}}{2} \right)}\right) + 1} & \text{otherwise} \end{cases}$$

/                                                                                                                                                                  /                True                  for x mod pi = 0
|                                                                                                                                                                  |                                                      
|                                                                                                                                                                  |/      True         for x mod pi = 0                  
|                                                                             1                                                                                for <|                                                     
|                                                                                                                                                                  |

$$\begin{cases} 1 & \text{for}\: \operatorname{ITE}\left(x \bmod \pi = 0, \text{True}, \operatorname{ITE}\left(x \bmod \pi = 0, \text{True}, \frac{\sin{\left(x \right)}}{2} \bmod \pi = 0\right)\right) \\\left(\left(\cot^{2}{\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}}{2} \right)}\right) - 1\right) \left(\begin{cases} 0 & \text{for}\: \operatorname{ITE}\left(x \bmod \pi = 0, \text{True}, \frac{\sin{\left(x \right)}}{2} \bmod \pi = 0\right) \\\left(- \frac{\cos{\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases} \right)}}{2}\right) + \frac{1}{2} & \text{otherwise} \end{cases}\right) & \text{otherwise} \end{cases}$$

/                                                                                        /         True           for x mod pi = 0
|                                                                                        |                                        
|                                                                                        |      /x\                               
|                                                                                        |   tan|-|                               
|                                        1                                           for <      \2/                               
|                                                                                        |----------- mod pi = 0     otherwise    
|                                                                                        |       2/x\                             
|                                                                                        |1 + tan |-|                             
|                                                                                        \        \2/                             
|                                                                                                                                 
|      //     0       for x mod pi = 0\                                                                                           
|      ||                             |                                                                                           
|      ||       /x\                   |                                                                                           
|      ||  2*tan|-|                   |                                                                                           
|      |<       \2/                   |                                                                                           
|      ||-----------     otherwise    |                                                                                           
|      ||       2/x\                  |                                                                                           
|      ||1 + tan |-|                  |                                                                                           
|     2|\        \2/                  | /                      1                  \                                               
|4*tan |------------------------------|*|-1 + ------------------------------------|                                               
|      \              4               / |         //     0       for x mod pi = 0\|                                               
|                                       |         ||                             ||                                               
<                                       |         ||       /x\                   ||                                               
|                                       |         ||  2*tan|-|                   ||                                               
|                                       |         |<       \2/                   ||                                               
|                                       |         ||-----------     otherwise    ||                                               
|                                       |         ||       2/x\                  ||                                               
|                                       |         ||1 + tan |-|                  ||                                               
|                                       |        2|\        \2/                  ||                                               
|                                       |     tan |------------------------------||                                               
|                                       \         \              2               //                                               
|----------------------------------------------------------------------------------                    otherwise                  
|                                                             2                                                                   
|                   /        //     0       for x mod pi = 0\\                                                                    
|                   |        ||                             ||                                                                    
|                   |        ||       /x\                   ||                                                                    
|                   |        ||  2*tan|-|                   ||                                                                    
|                   |        |<       \2/                   ||                                                                    
|                   |        ||-----------     otherwise    ||                                                                    
|                   |        ||       2/x\                  ||                                                                    
|                   |        ||1 + tan |-|                  ||                                                                    
|                   |       2|\        \2/                  ||                                                                    
|                   |1 + tan |------------------------------||                                                                    
|                   \        \              4               //                                                                    
\

$$\begin{cases} 1 & \text{for}\: \operatorname{ITE}\left(x \bmod \pi = 0, \text{True}, \frac{\tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} \bmod \pi = 0\right) \\\frac{4 \left(-1 + \left(\frac{1}{\tan^{2}{\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}}{2} \right)}}\right)\right) \left(\tan^{2}{\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}}{4} \right)}\right)}{\left(\left(\tan^{2}{\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}}{4} \right)}\right) + 1\right)^{2}} & \text{otherwise} \end{cases}$$

        /               /pi   x\       \
        |            csc|-- - -|       |
       2|pi             \2    2/       |
    csc |-- - -------------------------|
        |2    /       2/pi   x\\       |
        |     |    csc |-- - -||       |
        |     |        \2    2/|    /x\|
        |     |1 + ------------|*csc|-||
        |     |         2/x\   |    \2/|
        |     |      csc |-|   |       |
        \     \          \2/   /       /
1 - ------------------------------------
             /          /x\       \     
             |       sec|-|       |     
            2|          \2/       |     
         csc |--------------------|     
             |/       2/x\\       |     
             ||    sec |-||       |     
             ||        \2/|    /x\|     
             ||1 + -------|*csc|-||     
             ||       2/x\|    \2/|     
             ||    csc |-||       |     
             \\        \2//       /     
----------------------------------------
        /               /pi   x\       \
        |            csc|-- - -|       |
       2|pi             \2    2/       |
    csc |-- - -------------------------|
        |2    /       2/pi   x\\       |
        |     |    csc |-- - -||       |
        |     |        \2    2/|    /x\|
        |     |1 + ------------|*csc|-||
        |     |         2/x\   |    \2/|
        |     |      csc |-|   |       |
        \     \          \2/   /       /
1 + ------------------------------------
             /          /x\       \     
             |       sec|-|       |     
            2|          \2/       |     
         csc |--------------------|     
             |/       2/x\\       |     
             ||    sec |-||       |     
             ||        \2/|    /x\|     
             ||1 + -------|*csc|-||     
             ||       2/x\|    \2/|     
             ||    csc |-||       |     
             \\        \2//       /

$$\frac{1 - \frac{\csc^{2}{\left(\frac{\pi}{2} - \frac{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right) \csc{\left(\frac{x}{2} \right)}} \right)}}{\csc^{2}{\left(\frac{\sec{\left(\frac{x}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right) \csc{\left(\frac{x}{2} \right)}} \right)}}}{1 + \frac{\csc^{2}{\left(\frac{\pi}{2} - \frac{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right) \csc{\left(\frac{x}{2} \right)}} \right)}}{\csc^{2}{\left(\frac{\sec{\left(\frac{x}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right) \csc{\left(\frac{x}{2} \right)}} \right)}}}$$

        /                 /x   pi\       \
        |              cos|- - --|       |
       2|  pi             \2   2 /       |
    cos |- -- + -------------------------|
        |  2    /       2/x   pi\\       |
        |       |    cos |- - --||       |
        |       |        \2   2 /|    /x\|
        |       |1 + ------------|*cos|-||
        |       |         2/x\   |    \2/|
        |       |      cos |-|   |       |
        \       \          \2/   /       /
1 - --------------------------------------
           /          /x   pi\       \    
           |       cos|- - --|       |    
          2|          \2   2 /       |    
       cos |-------------------------|    
           |/       2/x   pi\\       |    
           ||    cos |- - --||       |    
           ||        \2   2 /|    /x\|    
           ||1 + ------------|*cos|-||    
           ||         2/x\   |    \2/|    
           ||      cos |-|   |       |    
           \\          \2/   /       /    
------------------------------------------
        /                 /x   pi\       \
        |              cos|- - --|       |
       2|  pi             \2   2 /       |
    cos |- -- + -------------------------|
        |  2    /       2/x   pi\\       |
        |       |    cos |- - --||       |
        |       |        \2   2 /|    /x\|
        |       |1 + ------------|*cos|-||
        |       |         2/x\   |    \2/|
        |       |      cos |-|   |       |
        \       \          \2/   /       /
1 + --------------------------------------
           /          /x   pi\       \    
           |       cos|- - --|       |    
          2|          \2   2 /       |    
       cos |-------------------------|    
           |/       2/x   pi\\       |    
           ||    cos |- - --||       |    
           ||        \2   2 /|    /x\|    
           ||1 + ------------|*cos|-||    
           ||         2/x\   |    \2/|    
           ||      cos |-|   |       |    
           \\          \2/   /       /

$$\frac{1 - \frac{\cos^{2}{\left(- \frac{\pi}{2} + \frac{\cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right) \cos{\left(\frac{x}{2} \right)}} \right)}}{\cos^{2}{\left(\frac{\cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right) \cos{\left(\frac{x}{2} \right)}} \right)}}}{1 + \frac{\cos^{2}{\left(- \frac{\pi}{2} + \frac{\cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right) \cos{\left(\frac{x}{2} \right)}} \right)}}{\cos^{2}{\left(\frac{\cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right) \cos{\left(\frac{x}{2} \right)}} \right)}}}$$

/                                                                                                                                                                                     /                  True                     for x mod pi = 0
|                                                                                                                                                                                     |                                                           
|                                                                                                                                                                                     |/         True           for x mod pi = 0                  
|                                                                                                                                                                                     ||                                                          
|                                                                                                                                                                                     ||      /x\                                                 
|                                                                                       1                                                                                         for <|   cot|-|                                                 
|                                                                                                                                                                                     |<      \2/                                    otherwise    
|                                                                                                                                                                                     ||----------- mod pi = 0     otherwise                      
|                                                                                                                                                                                     ||       2/x\                                               
|                                                                                                                                                                                     ||1 + cot |-|                                               
|                                                                                                                                                                                     \\        \2/                                               
|                                                                                                                                                                                                                                                 
|                                                               //                                                                    /         True           for x mod pi = 0\                                                                  
|                                                               ||                                                                    |                                        |                                                                  
|                                                               ||                                                                    |      /x\                               |                                                                  
|                                                               ||                                                                    |   cot|-|                               |                                                                  
|                                                               ||                              0                                 for <      \2/                               |                                                                  
|                                                               ||                                                                    |----------- mod pi = 0     otherwise    |                                                                  
|                                                               ||                                                                    |       2/x\                             |                                                                  
|                                                               ||                                                                    |1 + cot |-|                             |                                                                  
|/         //              0                 for x mod pi = 0\\ ||                                                                    \        \2/                             |                                                                  
||         ||                                                || ||                                                                                                             |                                                                  
||         ||/     0       for x mod pi = 0                  || ||        //              0                 for x mod pi = 0\                                                  |                                                                  
||         |||                                               || ||        ||                                                |                                                  |                                                                  
||         |||       /x\                                     || ||        ||/     0       for x mod pi = 0                  |                                                  |                                                                  
<|         |<|  2*cot|-|                                     || ||        |||                                               |                                                  |                                                                  
||         ||<       \2/                        otherwise    || ||        |||       /x\                                     |                                                  |                                                                  
||         |||-----------     otherwise                      || ||        |<|  2*cot|-|                                     |                                                  |                                                                  
||         |||       2/x\                                    || ||        ||<       \2/                        otherwise    |                                                  |                                                                  
||         |||1 + cot |-|                                    || ||        |||-----------     otherwise                      |                                                  |                                                                  
||        2|\\        \2/                                    || ||        |||       2/x\                                    |                                                  |                                                                  
||-1 + cot |-------------------------------------------------||*|<        |||1 + cot |-|                                    |                                                  |                             otherwise                            
|\         \                        2                        // ||       2|\\        \2/                                    |                                                  |                                                                  
|                                                               ||  4*cot |-------------------------------------------------|                                                  |                                                                  
|                                                               ||        \                        4                        /                                                  |                                                                  
|                                                               ||--------------------------------------------------------------                    otherwise                  |                                                                  
|                                                               ||                                                             2                                               |                                                                  
|                                                               ||/        //              0                 for x mod pi = 0\\                                                |                                                                  
|                                                               |||        ||                                                ||                                                |                                                                  
|                                                               |||        ||/     0       for x mod pi = 0                  ||                                                |                                                                  
|                                                               |||        |||                                               ||                                                |                                                                  
|                                                               |||        |||       /x\                                     ||                                                |                                                                  
|                                                               |||        |<|  2*cot|-|                                     ||                                                |                                                                  
|                                                               |||        ||<       \2/                        otherwise    ||                                                |                                                                  
|                                                               |||        |||-----------     otherwise                      ||                                                |                                                                  
|                                                               |||        |||       2/x\                                    ||                                                |                                                                  
|                                                               |||        |||1 + cot |-|                                    ||                                                |                                                                  
|                                                               |||       2|\\        \2/                                    ||                                                |                                                                  
|                                                               |||1 + cot |-------------------------------------------------||                                                |                                                                  
|                                                               ||\        \                        4                        //                                                |                                                                  
\                                                               \\                                                                                                             /

$$\begin{cases} 1 & \text{for}\: \operatorname{ITE}\left(x \bmod \pi = 0, \text{True}, \operatorname{ITE}\left(x \bmod \pi = 0, \text{True}, \frac{\cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} \bmod \pi = 0\right)\right) \\\left(\left(\cot^{2}{\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}}{2} \right)}\right) - 1\right) \left(\begin{cases} 0 & \text{for}\: \operatorname{ITE}\left(x \bmod \pi = 0, \text{True}, \frac{\cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} \bmod \pi = 0\right) \\\frac{4 \left(\cot^{2}{\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}}{4} \right)}\right)}{\left(\left(\cot^{2}{\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}}{4} \right)}\right) + 1\right)^{2}} & \text{otherwise} \end{cases}\right) & \text{otherwise} \end{cases}$$

/                                                             /                True                  for x mod pi = 0
|                                                             |                                                      
|                                                             |          sin(x)                                      
|                                                             |------------------------- mod pi = 0     otherwise    
|                                                             |  /        2    \                                     
|                           1                             for <  |     sin (x) |    2/x\                             
|                                                             |2*|1 + ---------|*sin |-|                             
|                                                             |  |         4/x\|     \2/                             
|                                                             |  |    4*sin |-||                                     
|                                                             |  \          \2//                                     
|                                                             \                                                      
|                                                                                                                    
|          //           0             for x mod pi = 0\                                                              
|          ||                                         |                                                              
|          ||         sin(x)                          |                                                              
|          ||-----------------------     otherwise    |                                                              
|         2||/        2    \                          |                                                              
|      sin |<|     sin (x) |    2/x\                  |                                                              
|          |||1 + ---------|*sin |-|                  |                                                              
|          |||         4/x\|     \2/                  |                                                              
|          |||    4*sin |-||                          |                                                              
|          ||\          \2//                          |                                                              
|          \\                                         /                                                              
|-1 + --------------------------------------------------                                                             
|           //           0             for x mod pi = 0\                                                             
|           ||                                         |                                                             
|           ||         sin(x)                          |                                                             
|           ||-----------------------     otherwise    |                                                             
|           ||/        2    \                          |                                                             
|           |<|     sin (x) |    2/x\                  |                                                             
|           |||1 + ---------|*sin |-|                  |                                                             
<           |||         4/x\|     \2/                  |                                                             
|           |||    4*sin |-||                          |                                                             
|           ||\          \2//                          |                                                             
|          4|\                                         |                                                             
|     4*sin |------------------------------------------|                                                             
|           \                    2                     /                                                             
|-------------------------------------------------------                           otherwise                         
|          //           0             for x mod pi = 0\                                                              
|          ||                                         |                                                              
|          ||         sin(x)                          |                                                              
|          ||-----------------------     otherwise    |                                                              
|         2||/        2    \                          |                                                              
|      sin |<|     sin (x) |    2/x\                  |                                                              
|          |||1 + ---------|*sin |-|                  |                                                              
|          |||         4/x\|     \2/                  |                                                              
|          |||    4*sin |-||                          |                                                              
|          ||\          \2//                          |                                                              
|          \\                                         /                                                              
| 1 + --------------------------------------------------                                                             
|           //           0             for x mod pi = 0\                                                             
|           ||                                         |                                                             
|           ||         sin(x)                          |                                                             
|           ||-----------------------     otherwise    |                                                             
|           ||/        2    \                          |                                                             
|           |<|     sin (x) |    2/x\                  |                                                             
|           |||1 + ---------|*sin |-|                  |                                                             
|           |||         4/x\|     \2/                  |                                                             
|           |||    4*sin |-||                          |                                                             
|           ||\          \2//                          |                                                             
|          4|\                                         |                                                             
|     4*sin |------------------------------------------|                                                             
\           \                    2                     /

$$\begin{cases} 1 & \text{for}\: \operatorname{ITE}\left(x \bmod \pi = 0, \text{True}, \frac{\sin{\left(x \right)}}{2 \cdot \left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right) \sin^{2}{\left(\frac{x}{2} \right)}} \bmod \pi = 0\right) \\\frac{-1 + \left(\frac{\sin^{2}{\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{\sin{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right) \sin^{2}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases} \right)}}{4 \left(\sin^{4}{\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{\sin{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right) \sin^{2}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}}{2} \right)}\right)}\right)}{1 + \left(\frac{\sin^{2}{\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{\sin{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right) \sin^{2}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases} \right)}}{4 \left(\sin^{4}{\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{\sin{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right) \sin^{2}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}}{2} \right)}\right)}\right)} & \text{otherwise} \end{cases}$$

/                                                                  /                 True                    for x mod pi = 0
|                                                                  |                                                         
|                                                                  |           sin(x)                                        
|                                                                  |---------------------------- mod pi = 0     otherwise    
|                                                                  |             /        2    \                             
|                             1                                for <             |     sin (x) |                             
|                                                                  |(1 - cos(x))*|1 + ---------|                             
|                                                                  |             |         4/x\|                             
|                                                                  |             |    4*sin |-||                             
|                                                                  |             \          \2//                             
|                                                                  \                                                         
|                                                                                                                            
|          //             0                for x mod pi = 0\                                                                 
|          ||                                              |                                                                 
|          ||          2*sin(x)                            |                                                                 
|          ||----------------------------     otherwise    |                                                                 
|         2||             /        2    \                  |                                                                 
|      sin |<             |     sin (x) |                  |                                                                 
|          ||(1 - cos(x))*|1 + ---------|                  |                                                                 
|          ||             |         4/x\|                  |                                                                 
|          ||             |    4*sin |-||                  |                                                                 
|          ||             \          \2//                  |                                                                 
|          \\                                              /                                                                 
|-1 + -------------------------------------------------------                                                                
|           //             0                for x mod pi = 0\                                                                
|           ||                                              |                                                                
|           ||          2*sin(x)                            |                                                                
|           ||----------------------------     otherwise    |                                                                
|           ||             /        2    \                  |                                                                
|           |<             |     sin (x) |                  |                                                                
|           ||(1 - cos(x))*|1 + ---------|                  |                                                                
<           ||             |         4/x\|                  |                                                                
|           ||             |    4*sin |-||                  |                                                                
|           ||             \          \2//                  |                                                                
|          4|\                                              |                                                                
|     4*sin |-----------------------------------------------|                                                                
|           \                       2                       /                                                                
|------------------------------------------------------------                            otherwise                           
|         //             0                for x mod pi = 0\                                                                  
|         ||                                              |                                                                  
|         ||          2*sin(x)                            |                                                                  
|         ||----------------------------     otherwise    |                                                                  
|        2||             /        2    \                  |                                                                  
|     sin |<             |     sin (x) |                  |                                                                  
|         ||(1 - cos(x))*|1 + ---------|                  |                                                                  
|         ||             |         4/x\|                  |                                                                  
|         ||             |    4*sin |-||                  |                                                                  
|         ||             \          \2//                  |                                                                  
|         \\                                              /                                                                  
|1 + -------------------------------------------------------                                                                 
|          //             0                for x mod pi = 0\                                                                 
|          ||                                              |                                                                 
|          ||          2*sin(x)                            |                                                                 
|          ||----------------------------     otherwise    |                                                                 
|          ||             /        2    \                  |                                                                 
|          |<             |     sin (x) |                  |                                                                 
|          ||(1 - cos(x))*|1 + ---------|                  |                                                                 
|          ||             |         4/x\|                  |                                                                 
|          ||             |    4*sin |-||                  |                                                                 
|          ||             \          \2//                  |                                                                 
|         4|\                                              |                                                                 
|    4*sin |-----------------------------------------------|                                                                 
\          \                       2                       /

$$\begin{cases} 1 & \text{for}\: \operatorname{ITE}\left(x \bmod \pi = 0, \text{True}, \frac{\sin{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right) \left(- \cos{\left(x \right)} + 1\right)} \bmod \pi = 0\right) \\\frac{-1 + \left(\frac{\sin^{2}{\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \sin{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right) \left(- \cos{\left(x \right)} + 1\right)} & \text{otherwise} \end{cases} \right)}}{4 \left(\sin^{4}{\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \sin{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right) \left(- \cos{\left(x \right)} + 1\right)} & \text{otherwise} \end{cases}}{2} \right)}\right)}\right)}{1 + \left(\frac{\sin^{2}{\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \sin{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right) \left(- \cos{\left(x \right)} + 1\right)} & \text{otherwise} \end{cases} \right)}}{4 \left(\sin^{4}{\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \sin{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right) \left(- \cos{\left(x \right)} + 1\right)} & \text{otherwise} \end{cases}}{2} \right)}\right)}\right)} & \text{otherwise} \end{cases}$$

/                                                             /             True                for x mod pi = 0
|                                                             |                                                 
|                                                             |          /x\                                    
|                                                             |       csc|-|                                    
|                                                             |          \2/                                    
|                                                             |-------------------- mod pi = 0     otherwise    
|                           1                             for

$$\begin{cases} 1 & \text{for}\: \operatorname{ITE}\left(x \bmod \pi = 0, \text{True}, \frac{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right) \sec{\left(\frac{x}{2} \right)}} \bmod \pi = 0\right) \\\frac{\left(\frac{\sec^{2}{\left(\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{x}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}} + 1\right) \sec{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}}{2}\right) - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{x}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}} + 1\right) \sec{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}}{2} \right)}}\right) - 1}{\left(\frac{\sec^{2}{\left(\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{x}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}} + 1\right) \sec{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}}{2}\right) - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{x}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}} + 1\right) \sec{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}}{2} \right)}}\right) + 1} & \text{otherwise} \end{cases}$$

/                                                                       /             True                for x mod pi = 0
|                                                                       |                                                 
|                                                                       |          /x\                                    
|                                                                       |       csc|-|                                    
|                                                                       |          \2/                                    
|                                                                       |-------------------- mod pi = 0     otherwise    
|                                1                                  for

$$\begin{cases} 1 & \text{for}\: \operatorname{ITE}\left(x \bmod \pi = 0, \text{True}, \frac{\csc{\left(\frac{x}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} \bmod \pi = 0\right) \\\frac{-1 + \left(\frac{\csc^{2}{\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{x}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}} + 1\right) \sec{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}}{2} \right)}}{\csc^{2}{\left(\left(- \frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{x}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}}{2}\right) + \frac{\pi}{2} \right)}}\right)}{1 + \left(\frac{\csc^{2}{\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{x}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}} + 1\right) \sec{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}}{2} \right)}}{\csc^{2}{\left(\left(- \frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{x}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}}{2}\right) + \frac{\pi}{2} \right)}}\right)} & \text{otherwise} \end{cases}$$

/                                                                       /             True                for x mod pi = 0
|                                                                       |                                                 
|                                                                       |          /x\                                    
|                                                                       |       cos|-|                                    
|                                                                       |          \2/                                    
|                                                                       |-------------------- mod pi = 0     otherwise    
|                                1                                  for

$$\begin{cases} 1 & \text{for}\: \operatorname{ITE}\left(x \bmod \pi = 0, \text{True}, \frac{\cos{\left(\frac{x}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} \bmod \pi = 0\right) \\\frac{-1 + \left(\frac{\cos^{2}{\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{x}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}}{2} \right)}}{\cos^{2}{\left(\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{x}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}}{2}\right) - \frac{\pi}{2} \right)}}\right)}{1 + \left(\frac{\cos^{2}{\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{x}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}}{2} \right)}}{\cos^{2}{\left(\left(\frac{\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{x}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}}{2}\right) - \frac{\pi}{2} \right)}}\right)} & \text{otherwise} \end{cases}$$

Piecewise((1, ITE(Mod(x = pi, 0), True, Mod(cos(x/2)/((1 + cos(x/2)^2/cos(x/2 - pi/2)^2)*cos(x/2 - pi/2)) = pi, 0))), ((-1 + cos(Piecewise((0, Mod(x = pi, 0)), (2*cos(x/2)/((1 + cos(x/2)^2/cos(x/2 - pi/2)^2)*cos(x/2 - pi/2)), True))/2)^2/cos(Piecewise((0, Mod(x = pi, 0)), (2*cos(x/2)/((1 + cos(x/2)^2/cos(x/2 - pi/2)^2)*cos(x/2 - pi/2)), True))/2 - pi/2)^2)/(1 + cos(Piecewise((0, Mod(x = pi, 0)), (2*cos(x/2)/((1 + cos(x/2)^2/cos(x/2 - pi/2)^2)*cos(x/2 - pi/2)), True))/2)^2/cos(Piecewise((0, Mod(x = pi, 0)), (2*cos(x/2)/((1 + cos(x/2)^2/cos(x/2 - pi/2)^2)*cos(x/2 - pi/2)), True))/2 - pi/2)^2), True))

Другие калькуляторы

Как пользоваться?

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