Господин Экзамен

Lang:

Другие калькуляторы

Как пользоваться?
6*t^2+4*t-1 если t=-4
sin(x)*sin(k*x) если x=1/3
(10*p-7*q)*(7*q+10*p) если q=-4
-15*a+16*b+6+15*a-12*b если a=1/3
Разложить многочлен на множители:
a^3*b^3+64
-x^2-3*x+1
x^6*y^6+64
-x+x^2-3*x
Общий знаменатель:
x/(x^2+y^2)-(y*(x-y)^2)/x^4-y^4
5/(x-7)-2/x-3*x/(x^2-49)+21/(49-x^2)
(a-y/a-b-b-y/a+b)/1/a^2-b^2
cos(x)/(1-sin(x))+cos(x)/(1+sin(x))
Умножение столбиком:
287*30
15*6094
65*5894
740*120
Деление столбиком:
3321/2
3476/2
3612/2
3841/2
Раскрыть скобки в:
3*(2*a-5*b^2)-12*a^2
(10*(x+1))^2
2*(3*m-n)+6*(2*n-m)
(p^2-p+5)*(24*p^2+p-5)
Разложение числа на простые множители:
74437
2995
17325
3802
Производная:
1/3
График функции y =:
1/3
Интеграл d{x}:
1/3
Идентичные выражения
sin(x)*sin(k*x) если x= один / три
синус от (x) умножить на синус от (k умножить на x) если x равно 1 делить на 3
синус от (x) умножить на синус от (k умножить на x) если x равно один делить на три
sin(x)sin(kx) если x=1/3
sinxsinkx если x=1/3
sin(x)*sin(k*x) при x=1/3
sin(x)*sin(k*x) если x=1 разделить на 3
Похожие выражения
sinx*sin(k*x) если x=1/3

Упрощение выражений/ sin(x)*sin(k*x) если x=1/3

sin(x)sin(kx) если x=1/3

Решение

Вы ввели [src]

sin(x)*sin(k*x)

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

sin(x)*sin(k*x)

Подстановка условия [src]

sin(x)*sin(k*x) при x = 1/3

подставляем

sin(x)*sin(k*x)

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

sin(x)*sin(k*x)

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

переменные

x = 1/3

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

sin((1/3))*sin(k*(1/3))

$$\sin{\left((1/3) \right)} \sin{\left((1/3) k \right)}$$

sin(1/3)*sin(k*1/3)

$$\sin{\left(\frac{1}{3} \right)} \sin{\left(k \frac{1}{3} \right)}$$

            /k\
sin(1/3)*sin|-|
            \3/

$$\sin{\left(\frac{1}{3} \right)} \sin{\left(\frac{k}{3} \right)}$$

sin(1/3)*sin(k/3)

Численный ответ [src]

sin(x)*sin(k*x)

sin(x)*sin(k*x)

Степени [src]

 /   -I*x    I*x\ /   -I*k*x    I*k*x\ 
-\- e     + e   /*\- e       + e     / 
---------------------------------------
                   4

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

-(-exp(-i*x) + exp(i*x))*(-exp(-i*k*x) + exp(i*k*x))/4

Собрать выражение [src]

cos(-x + k*x)   cos(x + k*x)
------------- - ------------
      2              2

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

cos(-x + k*x)/2 - cos(x + k*x)/2

Тригонометрическая часть [src]

       1       
---------------
csc(x)*csc(k*x)

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

            1            
-------------------------
csc(pi - x)*csc(pi - k*x)

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

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

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

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

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

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

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

cos(x*(-1 + k))   cos(x*(1 + k))
--------------- - --------------
       2                2

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

        1                  1        
----------------- - ----------------
2*sec(x*(-1 + k))   2*sec(x*(1 + k))

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

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

$$4 \sin{\left(\frac{x}{2} \right)} \sin{\left(\frac{k x}{2} \right)} \cos{\left(\frac{x}{2} \right)} \cos{\left(\frac{k x}{2} \right)}$$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

//  0     for x mod pi = 0\ //   0      for k*x mod pi = 0\
|<                        |*|<                            |
\\sin(x)     otherwise    / \\sin(k*x)      otherwise     /

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

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

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

//     0       for x mod pi = 0\ //       0         for k*x mod pi = 0\
||                             | ||                                   |
|<   /    pi\                  |*|<   /  pi      \                    |
||cos|x - --|     otherwise    | ||cos|- -- + k*x|      otherwise     |
\\   \    2 /                  / \\   \  2       /                    /

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

//            0              for x mod pi = 0\ //              0                for k*x mod pi = 0\
||                                           | ||                                                 |
|

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

//         0            for x mod pi = 0\ //           0              for k*x mod pi = 0\
||                                      | ||                                            |
||         2                            | ||           2                                |
||--------------------     otherwise    | ||------------------------      otherwise     |
|

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

//              0                 for x mod pi = 0\ //                  0                     for k*x mod pi = 0\
||                                                | ||                                                          |
||                /x\                             | ||                   /k*x\                                  |
||           2*cos|-|                             | ||              2*cos|---|                                  |
||                \2/                             | ||                   \ 2 /                                  |
||------------------------------     otherwise    | ||--------------------------------------      otherwise     |
|

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

//              0                 for x mod pi = 0\ //                0                   for k*x mod pi = 0\
||                                                | ||                                                      |
||                /x\                             | ||                 /k*x\                                |
||           2*csc|-|                             | ||            2*csc|---|                                |
||                \2/                             | ||                 \ 2 /                                |
||------------------------------     otherwise    | ||----------------------------------      otherwise     |
|

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

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

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