Господин Экзамен

Lang:

Другие калькуляторы

Как пользоваться?
x*2^x-5*2^x-4*x+20 если x=-1/4
sin(x)^2*cos(x) если x=1/3
2*(sqrt(3*x+1)/2-atan(sqrt(2)*sqrt(3*x+1))/(2^(3/2)))/3 если x=1/2
x^2-2/9 если x=-1/4
Разложить многочлен на множители:
-x^2+7*x+8
64*b^2+128*b+64
4*t^2-12*t*k+9*k^2
27*x^2-12*y^2
Общий знаменатель:
(3*b+(60*b/3*b-5)-5)/(1+(9*b^2-33*b+20/30*b-9*b^2-25))+24
((x^2+12*x)/(x^2-36))+(36/(x^2-36))
3/a+(a-3)/(a+5)
cos(50*pi)/360-cos(40*pi)/360
Умножение столбиком:
98*54
38*26
77*11
40*15
Деление столбиком:
66229/8
110/2
497/4
357/2
Раскрыть скобки в:
(3*a-2)*(3*a+2)-(a-8)*(a+8)
5*q*4*(p^3*q^2)^2*3*p^2*q^3
(b+2)*(b^2-2*b+4)-b*(b-5)*(b+5)
(b-3)*(b+3)
Разложение числа на простые множители:
3723
168
2224
2356
Производная:
1/3
График функции y =:
1/3
Интеграл d{x}:
1/3
Идентичные выражения
sin(x)^ два *cos(x) если x= один / три
синус от (x) в квадрате умножить на косинус от (x) если x равно 1 делить на 3
синус от (x) в степени два умножить на косинус от (x) если x равно один делить на три
sin(x)2*cos(x) если x=1/3
sinx2*cosx если x=1/3
sin(x)²*cos(x) если x=1/3
sin(x) в степени 2*cos(x) если x=1/3
sin(x)^2cos(x) если x=1/3
sin(x)2cos(x) если x=1/3
sinx2cosx если x=1/3
sinx^2cosx если x=1/3
sin(x)^2*cos(x) при x=1/3
sin(x)^2*cos(x) если x=1 разделить на 3
Похожие выражения
3*(-cos(x)^3-sin(x)^3+2*cos(x)^2*sin(x)+2*sin(x)^2*cos(x)) если x=-1
sinx^2*cosx если x=1/3

Упрощение выражений/ sin(x)^2*cos(x) если x=1/3

sin(x)^2*cos(x) если x=1/3

Решение

Вы ввели [src]

   2          
sin (x)*cos(x)

$$\sin^{2}{\left(x \right)} \cos{\left(x \right)}$$

sin(x)^2*cos(x)

Подстановка условия [src]

sin(x)^2*cos(x) при x = 1/3

подставляем

   2          
sin (x)*cos(x)

$$\sin^{2}{\left(x \right)} \cos{\left(x \right)}$$

   2          
sin (x)*cos(x)

$$\sin^{2}{\left(x \right)} \cos{\left(x \right)}$$

переменные

x = 1/3

$$x = \frac{1}{3}$$

   2                  
sin ((1/3))*cos((1/3))

$$\sin^{2}{\left((1/3) \right)} \cos{\left((1/3) \right)}$$

   2              
sin (1/3)*cos(1/3)

$$\sin^{2}{\left(\frac{1}{3} \right)} \cos{\left(\frac{1}{3} \right)}$$

sin(1/3)^2*cos(1/3)

Численный ответ [src]

sin(x)^2*cos(x)

sin(x)^2*cos(x)

Степени [src]

                 2 / I*x    -I*x\ 
 /   -I*x    I*x\  |e      e    | 
-\- e     + e   / *|---- + -----| 
                   \ 2       2  / 
----------------------------------
                4

$$- \frac{\left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right) \left(e^{i x} - e^{- i x}\right)^{2}}{4}$$

-(-exp(-i*x) + exp(i*x))^2*(exp(i*x)/2 + exp(-i*x)/2)/4

Собрать выражение [src]

  cos(3*x)   cos(x)
- -------- + ------
     4         4

$$\frac{\cos{\left(x \right)}}{4} - \frac{\cos{\left(3 x \right)}}{4}$$

-cos(3*x)/4 + cos(x)/4

Тригонометрическая часть [src]

      1       
--------------
   2          
csc (x)*sec(x)

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

   2/    pi\       
cos |x - --|*cos(x)
    \    2 /

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

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

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

  cos(3*x)   cos(x)
- -------- + ------
     4         4

$$\frac{\cos{\left(x \right)}}{4} - \frac{\cos{\left(3 x \right)}}{4}$$

         1         
-------------------
          2/    pi\
sec(x)*sec |x - --|
           \    2 /

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

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

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

         1         
-------------------
          2/pi    \
sec(x)*sec |-- - x|
           \2     /

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

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

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

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

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

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

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

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

$$\frac{8 \sin^{2}{\left(\frac{x}{2} \right)} \cos^{4}{\left(\frac{x}{2} \right)} \cos{\left(x \right)}}{\cos{\left(x \right)} + 1}$$

     6/x\    2/x\ /       2/x\\
4*cos |-|*tan |-|*|1 - tan |-||
      \2/     \2/ \        \2//

$$4 \cdot \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \cos^{6}{\left(\frac{x}{2} \right)} \tan^{2}{\left(\frac{x}{2} \right)}$$

     4/x\    2/x\ /       2/x\\
4*cos |-|*sin |-|*|1 - tan |-||
      \2/     \2/ \        \2//

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

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

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

       2/x\    4/x\        4/x\    2/x\
- 4*cos |-|*sin |-| + 4*cos |-|*sin |-|
        \2/     \2/         \2/     \2/

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

     2/x\    2/x\ /   2/x\      2/x\\
4*cos |-|*sin |-|*|cos |-| - sin |-||
      \2/     \2/ \    \2/       \2//

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

     2/x\ /       2/x\\
4*tan |-|*|1 - tan |-||
      \2/ \        \2//
-----------------------
                  3    
     /       2/x\\     
     |1 + tan |-||     
     \        \2//

$$\frac{4 \cdot \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3}}$$

     /       1   \    
   4*|1 - -------|    
     |       2/x\|    
     |    cot |-||    
     \        \2//    
----------------------
             3        
/       1   \     2/x\
|1 + -------| *cot |-|
|       2/x\|      \2/
|    cot |-||         
\        \2//

$$\frac{4 \cdot \left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{3} \cot^{2}{\left(\frac{x}{2} \right)}}$$

               6                               
  /       2/x\\     12/x\    2/x\ /       2/x\\
4*|1 - tan |-|| *cos  |-|*tan |-|*|1 - tan |-||
  \        \4//       \4/     \2/ \        \2//

$$4 \left(- \tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{6} \cdot \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \cos^{12}{\left(\frac{x}{4} \right)} \tan^{2}{\left(\frac{x}{2} \right)}$$

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

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

  /         2/x\   \
  |      sec |-|   |
  |          \2/   |
4*|1 - ------------|
  |       2/x   pi\|
  |    sec |- - --||
  \        \2   2 //
--------------------
   4/x\    2/x   pi\
sec |-|*sec |- - --|
    \2/     \2   2 /

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

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

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

                        /         4/x\\
                        |    4*sin |-||
      4/x\    6/pi   x\ |          \2/|
16*sin |-|*sin |-- + -|*|1 - ---------|
       \2/     \2    2/ |        2    |
                        \     sin (x) /
---------------------------------------
                   2                   
                sin (x)

$$\frac{16 \left(- \frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right) \sin^{4}{\left(\frac{x}{2} \right)} \sin^{6}{\left(\frac{x}{2} + \frac{\pi}{2} \right)}}{\sin^{2}{\left(x \right)}}$$

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

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

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

$$\frac{8 \tan^{2}{\left(\frac{x}{2} \right)} \tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)}$$

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

$$\frac{8 \tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)} \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$

  /       2/pi   x\\
  |    csc |-- - -||
  |        \2    2/|
4*|1 - ------------|
  |         2/x\   |
  |      csc |-|   |
  \          \2/   /
--------------------
   2/x\    4/pi   x\
csc |-|*csc |-- - -|
    \2/     \2    2/

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

               6                      
  /       2/x\\     2/x\ /       2/x\\
4*|1 - tan |-|| *tan |-|*|1 - tan |-||
  \        \4//      \2/ \        \2//
--------------------------------------
                         6            
            /       2/x\\             
            |1 + tan |-||             
            \        \4//

$$\frac{4 \left(- \tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{6} \cdot \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{6}}$$

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

$$\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases}\right)$$

      2       4/x\ /   2           4/x\\
16*sin (x)*sin |-|*|sin (x) - 4*sin |-||
               \2/ \                \2//
----------------------------------------
                              3         
         /   2           4/x\\          
         |sin (x) + 4*sin |-||          
         \                \2//

$$\frac{16 \left(- 4 \sin^{4}{\left(\frac{x}{2} \right)} + \sin^{2}{\left(x \right)}\right) \sin^{4}{\left(\frac{x}{2} \right)} \sin^{2}{\left(x \right)}}{\left(4 \sin^{4}{\left(\frac{x}{2} \right)} + \sin^{2}{\left(x \right)}\right)^{3}}$$

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

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

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

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

           /         4/x\\
           |    4*sin |-||
      4/x\ |          \2/|
16*sin |-|*|1 - ---------|
       \2/ |        2    |
           \     sin (x) /
--------------------------
                3         
 /         4/x\\          
 |    4*sin |-||          
 |          \2/|     2    
 |1 + ---------| *sin (x) 
 |        2    |          
 \     sin (x) /

$$\frac{16 \left(- \frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right) \sin^{4}{\left(\frac{x}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{3} \sin^{2}{\left(x \right)}}$$

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

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

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

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

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

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

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

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

                        //                              x             \
                        ||           1              for - mod 2*pi = 0|
                        ||                              2             |
     2/x\ /       2/x\\ ||                                            |
4*tan |-|*|1 - tan |-||*|<              6                             |
      \2/ \        \2// ||/        2/x\\     12/x\                    |
                        |||-1 + cot |-|| *sin  |-|      otherwise     |
                        ||\         \4//       \4/                    |
                        \\                                            /

$$4 \cdot \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(\begin{cases} 1 & \text{for}\: \frac{x}{2} \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{6} \sin^{12}{\left(\frac{x}{4} \right)} & \text{otherwise} \end{cases}\right) \tan^{2}{\left(\frac{x}{2} \right)}$$

                              //                             /    3*pi\             \
                              ||           1             for |x + ----| mod 2*pi = 0|
//  1     for x mod 2*pi = 0\ ||                             \     2  /             |
|<                          |*|<                                                    |
\\cos(x)      otherwise     / ||       4/x\        2/x\                             |
                              ||- 4*cos |-| + 4*cos |-|           otherwise         |
                              \\        \2/         \2/                             /

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

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

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

            /         2/x\   \  
            |      sec |-|   |  
       2/x\ |          \2/   |  
  4*sec |-|*|1 - ------------|  
        \2/ |       2/x   pi\|  
            |    sec |- - --||  
            \        \2   2 //  
--------------------------------
                  3             
/         2/x\   \              
|      sec |-|   |              
|          \2/   |     2/x   pi\
|1 + ------------| *sec |- - --|
|       2/x   pi\|      \2   2 /
|    sec |- - --||              
\        \2   2 //

$$\frac{4 \left(- \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec^{2}{\left(\frac{x}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{3} \sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}$$

               /       2/x   pi\\
               |    cos |- - --||
     2/x   pi\ |        \2   2 /|
4*cos |- - --|*|1 - ------------|
      \2   2 / |         2/x\   |
               |      cos |-|   |
               \          \2/   /
---------------------------------
                     3           
   /       2/x   pi\\            
   |    cos |- - --||            
   |        \2   2 /|     2/x\   
   |1 + ------------| *cos |-|   
   |         2/x\   |      \2/   
   |      cos |-|   |            
   \          \2/   /

$$\frac{4 \cdot \left(1 - \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right) \cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{3} \cos^{2}{\left(\frac{x}{2} \right)}}$$

               /       2/pi   x\\
               |    csc |-- - -||
     2/pi   x\ |        \2    2/|
4*csc |-- - -|*|1 - ------------|
      \2    2/ |         2/x\   |
               |      csc |-|   |
               \          \2/   /
---------------------------------
                     3           
   /       2/pi   x\\            
   |    csc |-- - -||            
   |        \2    2/|     2/x\   
   |1 + ------------| *csc |-|   
   |         2/x\   |      \2/   
   |      csc |-|   |            
   \          \2/   /

$$\frac{4 \cdot \left(1 - \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right) \csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{3} \csc^{2}{\left(\frac{x}{2} \right)}}$$

                //                     x             \
                ||       1         for - mod 2*pi = 0|
                ||                     2             |
                ||                                   |
                ||              6                    |
  /       1   \ ||/        2/x\\                     |
4*|1 - -------|*|<|-1 + cot |-||                     |
  |       2/x\| ||\         \4//                     |
  |    cot |-|| ||---------------      otherwise     |
  \        \2// ||              6                    |
                || /       2/x\\                     |
                || |1 + cot |-||                     |
                \\ \        \4//                     /
------------------------------------------------------
                          2/x\                        
                       cot |-|                        
                           \2/

$$\frac{4 \cdot \left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right) \left(\begin{cases} 1 & \text{for}\: \frac{x}{2} \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{6}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases}\right)}{\cot^{2}{\left(\frac{x}{2} \right)}}$$

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

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

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

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

//            0               for x mod pi = 0\                                                   
||                                            | //             1               for x mod 2*pi = 0\
||/   0     for x mod pi = 0                  | ||                                               |
|<|                                           |*|

$$\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Piecewise((0, Mod(x = pi, 0)), (4*csc(x/2)^2/((1 + csc(x/2)^2/csc(pi/2 - x/2)^2)^2*csc(pi/2 - x/2)^2), True))*Piecewise((1, Mod(x = 2*pi, 0)), ((-1 + csc(x/2)^2/csc(pi/2 - x/2)^2)/(1 + csc(x/2)^2/csc(pi/2 - x/2)^2), True))

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