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Другие калькуляторы

Как пользоваться?
-4*a^2+24*a-36 если a=-1/3
15*m+720*m-14*m если m=1/3
sin(p-a)*sin(p+a) если a=-1/3
(b-3)*(b-4)-(b+4)*2 если b=4
Разложить многочлен на множители:
w^2*w+7+14*w+49*w+7
a*h^7-a*m-u*h^7+n*m+u*m-n*h^7
x^6*y^9-64*x^3
x^3-y^3-1
Общий знаменатель:
m^2/(m-12)-24*m-144/(m-12)
2/x-4-x+8/x^2-16-1*x
cos(x-pi/2)-sin(x-pi/2)
(x-3/x^3-64+x-3/x^2+4*x+16)*2*x^3-128/3-x
Умножение столбиком:
79*97
86*38
92*20
97*28
Деление столбиком:
5316/6
5642/7
94023/991
56070/18917
Раскрыть скобки в:
3*(k-6)-(-27+k)
(x+y+z+1)*(x-1)*(y-1)*(z-1)
x*(x-3)*(x-4)
(c+d)*(d-c)
Разложение числа на простые множители:
8104
65283
69696
10011
Производная:
-1/3
Интеграл d{x}:
-1/3
Идентичные выражения
sin(p-a)*sin(p+a) если a=- один / три
синус от (p минус a) умножить на синус от (p плюс a) если a равно минус 1 делить на 3
синус от (p минус a) умножить на синус от (p плюс a) если a равно минус один делить на три
sin(p-a)sin(p+a) если a=-1/3
sinp-asinp+a если a=-1/3
sin(p-a)*sin(p+a) при a=-1/3
sin(p-a)*sin(p+a) если a=-1 разделить на 3
Похожие выражения
sin(p-a)*sin(p+a) если a=+1/3
sin(p-a)*sin(p-a) если a=-1/3
sin(p+a)*sin(p+a) если a=-1/3

Упрощение выражений/ sin(p-a)*sin(p+a) если a=-1/3

sin(p-a)*sin(p+a) если a=-1/3

Решение

Вы ввели [src]

sin(p - a)*sin(p + a)

$$\sin{\left(- a + p \right)} \sin{\left(a + p \right)}$$

sin(p - a)*sin(p + a)

Общее упрощение [src]

-sin(a + p)*sin(a - p)

$$- \sin{\left(a - p \right)} \sin{\left(a + p \right)}$$

-sin(a + p)*sin(a - p)

Подстановка условия [src]

sin(p - a)*sin(p + a) при a = -1/3

подставляем

sin(p - a)*sin(p + a)

$$\sin{\left(- a + p \right)} \sin{\left(a + p \right)}$$

-sin(a + p)*sin(a - p)

$$- \sin{\left(a - p \right)} \sin{\left(a + p \right)}$$

переменные

a = -1/3

$$a = - \frac{1}{3}$$

-sin((-1/3) + p)*sin((-1/3) - p)

$$- \sin{\left((-1/3) - p \right)} \sin{\left((-1/3) + p \right)}$$

-sin(-1/3 + p)*sin(-1/3 - p)

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

sin(-1/3 + p)*sin(1/3 + p)

$$\sin{\left(p - \frac{1}{3} \right)} \sin{\left(p + \frac{1}{3} \right)}$$

sin(-1/3 + p)*sin(1/3 + p)

Раскрыть выражение [src]

   2       2         2       2   
cos (a)*sin (p) - cos (p)*sin (a)

$$- \sin^{2}{\left(a \right)} \cos^{2}{\left(p \right)} + \sin^{2}{\left(p \right)} \cos^{2}{\left(a \right)}$$

(cos(a)*sin(p) + cos(p)*sin(a))*(cos(a)*sin(p) - cos(p)*sin(a))

$$\left(- \sin{\left(a \right)} \cos{\left(p \right)} + \sin{\left(p \right)} \cos{\left(a \right)}\right) \left(\sin{\left(a \right)} \cos{\left(p \right)} + \sin{\left(p \right)} \cos{\left(a \right)}\right)$$

(cos(a)*sin(p) + cos(p)*sin(a))*(cos(a)*sin(p) - cos(p)*sin(a))

Собрать выражение [src]

cos(2*a)   cos(2*p)
-------- - --------
   2          2

$$\frac{\cos{\left(2 a \right)}}{2} - \frac{\cos{\left(2 p \right)}}{2}$$

cos(2*a)/2 - cos(2*p)/2

Численный ответ [src]

sin(p + a)*sin(p - a)

sin(p + a)*sin(p - a)

Объединение рациональных выражений [src]

-sin(a + p)*sin(a - p)

$$- \sin{\left(a - p \right)} \sin{\left(a + p \right)}$$

-sin(a + p)*sin(a - p)

Рациональный знаменатель [src]

-sin(a + p)*sin(a - p)

$$- \sin{\left(a - p \right)} \sin{\left(a + p \right)}$$

-sin(a + p)*sin(a - p)

Степени [src]

-sin(a + p)*sin(a - p)

$$- \sin{\left(a - p \right)} \sin{\left(a + p \right)}$$

 /   I*(a - p)    I*(p - a)\ /   I*(-a - p)    I*(a + p)\ 
-\- e          + e         /*\- e           + e         / 
----------------------------------------------------------
                            4

$$- \frac{\left(- e^{i \left(- a - p\right)} + e^{i \left(a + p\right)}\right) \left(e^{i \left(- a + p\right)} - e^{i \left(a - p\right)}\right)}{4}$$

-(-exp(i*(a - p)) + exp(i*(p - a)))*(-exp(i*(-a - p)) + exp(i*(a + p)))/4

Комбинаторика [src]

-sin(a + p)*sin(a - p)

$$- \sin{\left(a - p \right)} \sin{\left(a + p \right)}$$

-sin(a + p)*sin(a - p)

Общий знаменатель [src]

-sin(a + p)*sin(a - p)

$$- \sin{\left(a - p \right)} \sin{\left(a + p \right)}$$

-sin(a + p)*sin(a - p)

Тригонометрическая часть [src]

-sin(a + p)*sin(a - p)

$$- \sin{\left(a - p \right)} \sin{\left(a + p \right)}$$

         -1          
---------------------
csc(a + p)*csc(a - p)

$$- \frac{1}{\csc{\left(a - p \right)} \csc{\left(a + p \right)}}$$

              -1               
-------------------------------
csc(pi + p - a)*csc(pi - a - p)

$$- \frac{1}{\csc{\left(- a - p + \pi \right)} \csc{\left(- a + p + \pi \right)}}$$

    /        pi\    /        pi\
-cos|a + p - --|*cos|a - p - --|
    \        2 /    \        2 /

$$- \cos{\left(a - p - \frac{\pi}{2} \right)} \cos{\left(a + p - \frac{\pi}{2} \right)}$$

              -1               
-------------------------------
   /        pi\    /        pi\
sec|a + p - --|*sec|a - p - --|
   \        2 /    \        2 /

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

              -1               
-------------------------------
   /    pi    \    /pi        \
sec|p + -- - a|*sec|-- - a - p|
   \    2     /    \2         /

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

      2/a + p\    2/a - p\    /a   p\    /a   p\
-4*sin |-----|*sin |-----|*cot|- + -|*cot|- - -|
       \  2  /     \  2  /    \2   2/    \2   2/

$$- 4 \sin^{2}{\left(\frac{a - p}{2} \right)} \sin^{2}{\left(\frac{a + p}{2} \right)} \cot{\left(\frac{a}{2} - \frac{p}{2} \right)} \cot{\left(\frac{a}{2} + \frac{p}{2} \right)}$$

            /a   p\    /a   p\     
      -4*cot|- + -|*cot|- - -|     
            \2   2/    \2   2/     
-----------------------------------
/       2/a   p\\ /       2/a   p\\
|1 + cot |- + -||*|1 + cot |- - -||
\        \2   2// \        \2   2//

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

            /a   p\    /a   p\     
      -4*tan|- + -|*tan|- - -|     
            \2   2/    \2   2/     
-----------------------------------
/       2/a   p\\ /       2/a   p\\
|1 + tan |- + -||*|1 + tan |- - -||
\        \2   2// \        \2   2//

$$- \frac{4 \tan{\left(\frac{a}{2} - \frac{p}{2} \right)} \tan{\left(\frac{a}{2} + \frac{p}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} - \frac{p}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{a}{2} + \frac{p}{2} \right)} + 1\right)}$$

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

$$- \frac{\left(\cos{\left(a - p \right)} + 1\right) \left(\cos{\left(a + p \right)} + 1\right) \sin{\left(\frac{a - p}{2} \right)} \sin{\left(\frac{a + p}{2} \right)}}{\cos{\left(\frac{a - p}{2} \right)} \cos{\left(\frac{a + p}{2} \right)}}$$

                                      /a   p\    /a   p\    /a + p\    /a - p\
-(1 + cos(a + p))*(1 + cos(a - p))*sec|- + -|*sec|- - -|*sin|-----|*sin|-----|
                                      \2   2/    \2   2/    \  2  /    \  2  /

$$- \left(\cos{\left(a - p \right)} + 1\right) \left(\cos{\left(a + p \right)} + 1\right) \sin{\left(\frac{a - p}{2} \right)} \sin{\left(\frac{a + p}{2} \right)} \sec{\left(\frac{a}{2} - \frac{p}{2} \right)} \sec{\left(\frac{a}{2} + \frac{p}{2} \right)}$$

                           -4                            
---------------------------------------------------------
/         1     \ /         1     \    /a   p\    /a   p\
|1 + -----------|*|1 + -----------|*cot|- + -|*cot|- - -|
|       2/a   p\| |       2/a   p\|    \2   2/    \2   2/
|    cot |- + -|| |    cot |- - -||                      
\        \2   2// \        \2   2//

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

 //    0       for (a + p) mod pi = 0\ //    0       for (a - p) mod pi = 0\
-|<                                  |*|<                                  |
 \\sin(a + p)        otherwise       / \\sin(a - p)        otherwise       /

$$- \left(\begin{cases} 0 & \text{for}\: \left(a - p\right) \bmod \pi = 0 \\\sin{\left(a - p \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(a + p\right) \bmod \pi = 0 \\\sin{\left(a + p \right)} & \text{otherwise} \end{cases}\right)$$

 //    0       for (a + p) mod pi = 0\ //    0       for (a - p) mod pi = 0\
 ||                                  | ||                                  |
-|<    1                             |*|<    1                             |
 ||----------        otherwise       | ||----------        otherwise       |
 \\csc(a + p)                        / \\csc(a - p)                        /

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

 //       0         for (a + p) mod pi = 0\ //       0         for (a - p) mod pi = 0\
 ||                                       | ||                                       |
-|<   /        pi\                        |*|<   /        pi\                        |
 ||cos|a + p - --|        otherwise       | ||cos|a - p - --|        otherwise       |
 \\   \        2 /                        / \\   \        2 /                        /

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

                  /        2/p   a   pi\\ /        2/a   p   pi\\ 
-(1 - sin(a - p))*|-1 + cot |- - - + --||*|-1 + tan |- + - + --|| 
                  \         \2   2   4 // \         \2   2   4 // 
------------------------------------------------------------------
                       /       2/a   p   pi\\                     
                     2*|1 + tan |- + - + --||                     
                       \        \2   2   4 //

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

 /       2/a   p   pi\\ /       2/p   a   pi\\                  
-|1 - cot |- + - + --||*|1 - tan |- - - + --||*(1 + sin(a + p)) 
 \        \2   2   4 // \        \2   2   4 //                  
----------------------------------------------------------------
                      /       2/p   a   pi\\                    
                    2*|1 + tan |- - - + --||                    
                      \        \2   2   4 //

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

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

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

                        2/a + p\    2/a - p\                 
                 -16*sin |-----|*sin |-----|                 
                         \  2  /     \  2  /                 
-------------------------------------------------------------
/         4/a + p\\ /         4/a - p\\                      
|    4*sin |-----|| |    4*sin |-----||                      
|          \  2  /| |          \  2  /|                      
|1 + -------------|*|1 + -------------|*sin(a + p)*sin(a - p)
|        2        | |        2        |                      
\     sin (a + p) / \     sin (a - p) /

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

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

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

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

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

                        2/a   p\    2/a   p\                 
                 -16*sin |- + -|*sin |- - -|                 
                         \2   2/     \2   2/                 
-------------------------------------------------------------
/         4/a   p\\ /         4/a   p\\                      
|    4*sin |- + -|| |    4*sin |- - -||                      
|          \2   2/| |          \2   2/|                      
|1 + -------------|*|1 + -------------|*sin(a + p)*sin(a - p)
|        2        | |        2        |                      
\     sin (a + p) / \     sin (a - p) /

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

 //      0         for (a + p) mod pi = 0\ //      0         for (a - p) mod pi = 0\
 ||                                      | ||                                      |
 ||1 - cos(a + p)                        | ||1 - cos(a - p)                        |
-|<--------------        otherwise       |*|<--------------        otherwise       |
 ||     /a   p\                          | ||     /a   p\                          |
 ||  tan|- + -|                          | ||  tan|- - -|                          |
 \\     \2   2/                          / \\     \2   2/                          /

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

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

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

 //       0         for (a + p) mod pi = 0\ //       0         for (a - p) mod pi = 0\
 ||                                       | ||                                       |
 ||       /a   p\                         | ||       /a   p\                         |
 ||  2*tan|- + -|                         | ||  2*tan|- - -|                         |
-|<       \2   2/                         |*|<       \2   2/                         |
 ||---------------        otherwise       | ||---------------        otherwise       |
 ||       2/a   p\                        | ||       2/a   p\                        |
 ||1 + tan |- + -|                        | ||1 + tan |- - -|                        |
 \\        \2   2/                        / \\        \2   2/                        /

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

 //                 0                   for (a + p) mod pi = 0\ //                 0                   for (a - p) mod pi = 0\
 ||                                                           | ||                                                           |
-|

$$- \left(\begin{cases} 0 & \text{for}\: \left(a - p\right) \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \left(a - p\right) \bmod \pi = 0 \\\sin{\left(a - p \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(a + p\right) \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \left(a + p\right) \bmod \pi = 0 \\\sin{\left(a + p \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$

 //             0                for (a + p) mod pi = 0\ //             0                for (a - p) mod pi = 0\
 ||                                                    | ||                                                    |
 ||             2                                      | ||             2                                      |
 ||----------------------------        otherwise       | ||----------------------------        otherwise       |
-|

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

                       /a   p   pi\    /a   pi   p\                
                 -4*cos|- + - - --|*cos|- - -- - -|                
                       \2   2   2 /    \2   2    2/                
-------------------------------------------------------------------
/       2/a   p   pi\\ /       2/a   pi   p\\                      
|    cos |- + - - --|| |    cos |- - -- - -||                      
|        \2   2   2 /| |        \2   2    2/|    /a   p\    /a   p\
|1 + ----------------|*|1 + ----------------|*cos|- + -|*cos|- - -|
|         2/a   p\   | |         2/a   p\   |    \2   2/    \2   2/
|      cos |- + -|   | |      cos |- - -|   |                      
\          \2   2/   / \          \2   2/   /

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

                                 /a   p\    /a   p\                          
                           -4*sec|- + -|*sec|- - -|                          
                                 \2   2/    \2   2/                          
-----------------------------------------------------------------------------
/         2/a   p\   \ /         2/a   p\   \                                
|      sec |- + -|   | |      sec |- - -|   |                                
|          \2   2/   | |          \2   2/   |    /a   p   pi\    /a   pi   p\
|1 + ----------------|*|1 + ----------------|*sec|- + - - --|*sec|- - -- - -|
|       2/a   p   pi\| |       2/a   pi   p\|    \2   2   2 /    \2   2    2/
|    sec |- + - - --|| |    sec |- - -- - -||                                
\        \2   2   2 // \        \2   2    2//

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

                       /pi   p   a\    /pi   a   p\                
                 -4*csc|-- + - - -|*csc|-- - - - -|                
                       \2    2   2/    \2    2   2/                
-------------------------------------------------------------------
/       2/pi   a   p\\ /       2/pi   p   a\\                      
|    csc |-- - - - -|| |    csc |-- + - - -||                      
|        \2    2   2/| |        \2    2   2/|    /a   p\    /a   p\
|1 + ----------------|*|1 + ----------------|*csc|- + -|*csc|- - -|
|         2/a   p\   | |         2/a   p\   |    \2   2/    \2   2/
|      csc |- + -|   | |      csc |- - -|   |                      
\          \2   2/   / \          \2   2/   /

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

 //                           /        3*pi\             \                                                                             
 ||          1            for |a + p + ----| mod 2*pi = 0| //                                              /        3*pi\             \
 ||                           \         2  /             | ||                   1                      for |a - p + ----| mod 2*pi = 0|
 ||                                                      | ||                                              \         2  /             |
 ||        2/a   p   pi\                                 | ||                                                                         |
-|<-1 + tan |- + - + --|                                 |*|<                 /        2/p   a   pi\\                                 |
 ||         \2   2   4 /                                 | ||(1 - sin(a - p))*|-1 + cot |- - - + --||                                 |
 ||---------------------             otherwise           | ||                 \         \2   2   4 //                                 |
 ||        2/a   p   pi\                                 | ||----------------------------------------             otherwise           |
 || 1 + tan |- + - + --|                                 | \\                   2                                                     /
 \\         \2   2   4 /                                 /

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

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

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

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

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

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

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

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

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

 //                0                  for (a + p) mod pi = 0\ //                0                  for (a - p) mod pi = 0\
 ||                                                         | ||                                                         |
 ||             /a   p   pi\                                | ||             /a   pi   p\                                |
 ||        2*sec|- + - - --|                                | ||        2*sec|- - -- - -|                                |
 ||             \2   2   2 /                                | ||             \2   2    2/                                |
 ||---------------------------------        otherwise       | ||---------------------------------        otherwise       |
-|

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

 //                  0                     for (a + p) mod pi = 0\ //                  0                     for (a - p) mod pi = 0\
 ||                                                              | ||                                                              |
 ||                  /a   p\                                     | ||                  /a   p\                                     |
 ||             2*cos|- + -|                                     | ||             2*cos|- - -|                                     |
 ||                  \2   2/                                     | ||                  \2   2/                                     |
 ||--------------------------------------        otherwise       | ||--------------------------------------        otherwise       |
-|

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

 //                  0                     for (a + p) mod pi = 0\ //                  0                     for (a - p) mod pi = 0\
 ||                                                              | ||                                                              |
 ||                  /a   p\                                     | ||                  /a   p\                                     |
 ||             2*csc|- + -|                                     | ||             2*csc|- - -|                                     |
 ||                  \2   2/                                     | ||                  \2   2/                                     |
 ||--------------------------------------        otherwise       | ||--------------------------------------        otherwise       |
-|

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

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

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