Господин Экзамен
Другие калькуляторы
-
Как пользоваться?
- cos(a)^2-cos(a)^4+sin(a)^4 если a=-4
- x-x-x-(-1)*x+3 если x=-1/2
- 7*cos(x)^2-5+7*sin(x)^2 если x=2
- 32*a+8*b если a=2
- Разложить многочлен на множители:
- a^3-b^15*c^18
- c^3+d^3
- 144*a^2*b^4-24*a^3*b^3
- y^3-y
- Общий знаменатель:
- t/(k-9)+18/(9-k)
- (sqrt(m)/(m-16)-sqrt(m)/((4-sqrt(m))^2))*m^2-8*m+16/(4*sqrt(m))+2/sqrt(m)+4
- tan(pi/4+a/2)*((1-sin(a))/cos(a))
- (3*q^3-81*p^3)/(18*p^2*q+16*x-16)+(81*q^2*p-54*q*p^2+9*p^3)/(2*q*p^2-12*p*q^2+18*q^3)
- Умножение столбиком:
- 816*209
- 15*14
- 30*14
- 26*35
- Деление столбиком:
- 2863/7
- 72/8
- 32068/4
- 540/36
- Раскрыть скобки в:
- (x-12)*(x+8)
- (2*a+1)*(5*a-6)
- (p+3)*(p-11)+(p+6)^2
- (m+5)^2-(m-4)*(m+4)
- Разложение числа на простые множители:
- 2023
- 1224
- 108
- 2024
- Производная:
- -4
- Интеграл d{x}:
-
-4
- График функции y =:
- -4
-
Идентичные выражения
- cos(a)^ два -cos(a)^ четыре +sin(a)^ четыре если a=- четыре
- косинус от (a) в квадрате минус косинус от (a) в степени 4 плюс синус от (a) в степени 4 если a равно минус 4
- косинус от (a) в степени два минус косинус от (a) в степени четыре плюс синус от (a) в степени четыре если a равно минус четыре
- cos(a)2-cos(a)4+sin(a)4 если a=-4
- cosa2-cosa4+sina4 если a=-4
- cos(a)²-cos(a)⁴+sin(a)⁴ если a=-4
- cos(a) в степени 2-cos(a) в степени 4+sin(a) в степени 4 если a=-4
- cosa^2-cosa^4+sina^4 если a=-4
- cos(a)^2-cos(a)^4+sin(a)^4 при a=-4
-
Похожие выражения
- cos(a)^2+cos(a)^4+sin(a)^4 если a=-4
- cos(a)^2-cos(a)^4+sin(a)^4 если a=+4
- cos(a)^2-cos(a)^4-sin(a)^4 если a=-4
cos(a)^2-cos(a)^4+sin(a)^4 если a=-4
Решение
Вы ввели
[src]
2 4 4 cos (a) - cos (a) + sin (a)
$$\sin^{4}{\left(a \right)} - \cos^{4}{\left(a \right)} + \cos^{2}{\left(a \right)}$$
cos(a)^2 - cos(a)^4 + sin(a)^4
Подстановка условия
[src]
cos(a)^2 - cos(a)^4 + sin(a)^4 при a = -4
подставляем
2 4 4 cos (a) - cos (a) + sin (a)
$$\sin^{4}{\left(a \right)} - \cos^{4}{\left(a \right)} + \cos^{2}{\left(a \right)}$$
2 sin (a)
$$\sin^{2}{\left(a \right)}$$
переменные
a = -4
$$a = -4$$
2 sin ((-4))
$$\sin^{2}{\left((-4) \right)}$$
2 sin (-4)
$$\sin^{2}{\left(-4 \right)}$$
2 sin (4)
$$\sin^{2}{\left(4 \right)}$$
sin(4)^2
Степени
[src]
2 4 4 / I*a -I*a\ / I*a -I*a\ / -I*a I*a\ |e e | |e e | \- e + e / |---- + -----| - |---- + -----| + ----------------- \ 2 2 / \ 2 2 / 16
$$- \left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2}\right)^{4} + \frac{\left(e^{i a} - e^{- i a}\right)^{4}}{16} + \left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2}\right)^{2}$$
(exp(i*a)/2 + exp(-i*a)/2)^2 - (exp(i*a)/2 + exp(-i*a)/2)^4 + (-exp(-i*a) + exp(i*a))^4/16
Собрать выражение
[src]
1 cos(2*a) - - -------- 2 2
$$- \frac{\cos{\left(2 a \right)}}{2} + \frac{1}{2}$$
1/2 - cos(2*a)/2
Тригонометрическая часть
[src]
2 sin (a)
$$\sin^{2}{\left(a \right)}$$
1 ------- 2 csc (a)
$$\frac{1}{\csc^{2}{\left(a \right)}}$$
2/ pi\
cos |a - --|
\ 2 /
$$\cos^{2}{\left(a - \frac{\pi}{2} \right)}$$
1 cos(2*a) - - -------- 2 2
$$- \frac{\cos{\left(2 a \right)}}{2} + \frac{1}{2}$$
1
------------
2/ pi\
sec |a - --|
\ 2 /
$$\frac{1}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}$$
2 2 1 sin (a) cos (a) - + ------- - ------- 2 2 2
$$\frac{\sin^{2}{\left(a \right)}}{2} - \frac{\cos^{2}{\left(a \right)}}{2} + \frac{1}{2}$$
2 4/ pi\ 4
cos (a) + cos |a - --| - cos (a)
\ 2 /
$$- \cos^{4}{\left(a \right)} + \cos^{4}{\left(a - \frac{\pi}{2} \right)} + \cos^{2}{\left(a \right)}$$
2/a\
4*tan |-|
\2/
--------------
2
/ 2/a\\
|1 + tan |-||
\ \2//
$$\frac{4 \tan^{2}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}$$
/ 0 for a mod pi = 0 | < 2 |sin (a) otherwise \
$$\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin^{2}{\left(a \right)} & \text{otherwise} \end{cases}$$
1 1 1 ------- + ------- - ------- 4 2 4 csc (a) sec (a) sec (a)
$$\frac{1}{\sec^{2}{\left(a \right)}} - \frac{1}{\sec^{4}{\left(a \right)}} + \frac{1}{\csc^{4}{\left(a \right)}}$$
4 2/ pi\ 4/ pi\
sin (a) + sin |a + --| - sin |a + --|
\ 2 / \ 2 /
$$\sin^{4}{\left(a \right)} - \sin^{4}{\left(a + \frac{\pi}{2} \right)} + \sin^{2}{\left(a + \frac{\pi}{2} \right)}$$
1 1 1
------- + ------------ - -------
2 4/ pi\ 4
sec (a) sec |a - --| sec (a)
\ 2 /
$$\frac{1}{\sec^{2}{\left(a \right)}} + \frac{1}{\sec^{4}{\left(a - \frac{\pi}{2} \right)}} - \frac{1}{\sec^{4}{\left(a \right)}}$$
1 1 1
------- + ------------ - -------
2 4/pi \ 4
sec (a) sec |-- - a| sec (a)
\2 /
$$\frac{1}{\sec^{2}{\left(a \right)}} + \frac{1}{\sec^{4}{\left(- a + \frac{\pi}{2} \right)}} - \frac{1}{\sec^{4}{\left(a \right)}}$$
2 4 4 4/a\
cos (a) - cos (a) + (1 + cos(a)) *tan |-|
\2/
$$\left(\cos{\left(a \right)} + 1\right)^{4} \tan^{4}{\left(\frac{a}{2} \right)} - \cos^{4}{\left(a \right)} + \cos^{2}{\left(a \right)}$$
1 1 1
------- + ------------ - ------------
4 2/pi \ 4/pi \
csc (a) csc |-- - a| csc |-- - a|
\2 / \2 /
$$\frac{1}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} - \frac{1}{\csc^{4}{\left(- a + \frac{\pi}{2} \right)}} + \frac{1}{\csc^{4}{\left(a \right)}}$$
2 4/a\
16*sin (a)*sin |-|
\2/
----------------------
2
/ 2 4/a\\
|sin (a) + 4*sin |-||
\ \2//
$$\frac{16 \sin^{4}{\left(\frac{a}{2} \right)} \sin^{2}{\left(a \right)}}{\left(4 \sin^{4}{\left(\frac{a}{2} \right)} + \sin^{2}{\left(a \right)}\right)^{2}}$$
1 1 1
------------ + ------------ - ------------
4 2/pi \ 4/pi \
csc (pi - a) csc |-- - a| csc |-- - a|
\2 / \2 /
$$\frac{1}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} - \frac{1}{\csc^{4}{\left(- a + \frac{\pi}{2} \right)}} + \frac{1}{\csc^{4}{\left(- a + \pi \right)}}$$
2 4 6/a\ 4/a\ 8/a\
sin (a) - sin (a) - 32*sin |-| + 16*sin |-| + 16*sin |-|
\2/ \2/ \2/
$$16 \sin^{8}{\left(\frac{a}{2} \right)} - 32 \sin^{6}{\left(\frac{a}{2} \right)} + 16 \sin^{4}{\left(\frac{a}{2} \right)} - \sin^{4}{\left(a \right)} + \sin^{2}{\left(a \right)}$$
/ 0 for a mod pi = 0 | | 2/a\ | 4*cot |-| | \2/ <-------------- otherwise | 2 |/ 2/a\\ ||1 + cot |-|| |\ \2// \
$$\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}$$
4
1 + cos(2*a) / 2/a\\ 8/a\ 4/a\ 4/a\
------------ - |1 - tan |-|| *cos |-| + 16*cos |-|*sin |-|
2 \ \2// \2/ \2/ \2/
$$- \left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4} \cos^{8}{\left(\frac{a}{2} \right)} + 16 \sin^{4}{\left(\frac{a}{2} \right)} \cos^{4}{\left(\frac{a}{2} \right)} + \frac{\cos{\left(2 a \right)} + 1}{2}$$
4
1 + cos(2*a) / 2/a\\ 8/a\ 8/a\ 4/a\
------------ - |1 - tan |-|| *cos |-| + 16*cos |-|*tan |-|
2 \ \2// \2/ \2/ \2/
$$- \left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4} \cos^{8}{\left(\frac{a}{2} \right)} + 16 \cos^{8}{\left(\frac{a}{2} \right)} \tan^{4}{\left(\frac{a}{2} \right)} + \frac{\cos{\left(2 a \right)} + 1}{2}$$
4 1 cos(2*a) / 2/a\\ 8/a\ 4/a\ 4/a\ - + -------- - |1 - tan |-|| *cos |-| + 16*cos |-|*sin |-| 2 2 \ \2// \2/ \2/ \2/
$$- \left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4} \cos^{8}{\left(\frac{a}{2} \right)} + 16 \sin^{4}{\left(\frac{a}{2} \right)} \cos^{4}{\left(\frac{a}{2} \right)} + \frac{\cos{\left(2 a \right)}}{2} + \frac{1}{2}$$
4/a\
4 16*tan |-|
1 + cos(2*a) / 2/a\\ 8/a\ \2/
------------ - |-1 + cot |-|| *sin |-| + --------------
2 \ \2// \2/ 4
/ 2/a\\
|1 + tan |-||
\ \2//
$$- \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{4} \sin^{8}{\left(\frac{a}{2} \right)} + \frac{\cos{\left(2 a \right)} + 1}{2} + \frac{16 \tan^{4}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4}}$$
4 8/a\
16*cos (a)*cos |-|
1 cos(2*a) 4/a\ 4/a\ \2/
- + -------- + 16*cos |-|*sin |-| - ------------------
2 2 \2/ \2/ 4
(1 + cos(a))
$$16 \sin^{4}{\left(\frac{a}{2} \right)} \cos^{4}{\left(\frac{a}{2} \right)} - \frac{16 \cos^{8}{\left(\frac{a}{2} \right)} \cos^{4}{\left(a \right)}}{\left(\cos{\left(a \right)} + 1\right)^{4}} + \frac{\cos{\left(2 a \right)}}{2} + \frac{1}{2}$$
4
/ 2/a pi\\ 4
4 |1 - cot |- + --|| *(1 + sin(a))
1 + cos(2*a) / 2/a\\ 8/a\ \ \2 4 //
------------ - |1 - tan |-|| *cos |-| + ---------------------------------
2 \ \2// \2/ 16
$$- \left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4} \cos^{8}{\left(\frac{a}{2} \right)} + \frac{\left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{4} \left(\sin{\left(a \right)} + 1\right)^{4}}{16} + \frac{\cos{\left(2 a \right)} + 1}{2}$$
4
/ 2/a pi\\
| cos |- - --||
1 cos(2*a) | \2 2 /| 8/a\ 4/a\ 4/a pi\
- + -------- - |1 - ------------| *cos |-| + 16*cos |-|*cos |- - --|
2 2 | 2/a\ | \2/ \2/ \2 2 /
| cos |-| |
\ \2/ /
$$- \left(1 - \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right)^{4} \cos^{8}{\left(\frac{a}{2} \right)} + 16 \cos^{4}{\left(\frac{a}{2} \right)} \cos^{4}{\left(\frac{a}{2} - \frac{\pi}{2} \right)} + \frac{\cos{\left(2 a \right)}}{2} + \frac{1}{2}$$
4
/ 2/a\ \
| sec |-| |
| \2/ |
|1 - ------------|
| 2/a pi\|
| sec |- - --||
1 1 \ \2 2 // 16
- + ---------- - ------------------- + --------------------
2 2*sec(2*a) 8/a\ 4/a\ 4/a pi\
sec |-| sec |-|*sec |- - --|
\2/ \2/ \2 2 /
$$\frac{1}{2} + \frac{1}{2 \sec{\left(2 a \right)}} - \frac{\left(- \frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)^{4}}{\sec^{8}{\left(\frac{a}{2} \right)}} + \frac{16}{\sec^{4}{\left(\frac{a}{2} \right)} \sec^{4}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}$$
4
/pi \ / 4/a\\ 8/a\ 8/pi a\
sin|-- + 2*a| | 4*sin |-|| 256*sin |-|*sin |-- + -|
1 \2 / | \2/| 8/pi a\ \2/ \2 2/
- + ------------- - |1 - ---------| *sin |-- + -| + ------------------------
2 2 | 2 | \2 2/ 4
\ sin (a) / sin (a)
$$- \left(- \frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right)^{4} \sin^{8}{\left(\frac{a}{2} + \frac{\pi}{2} \right)} + \frac{256 \sin^{8}{\left(\frac{a}{2} \right)} \sin^{8}{\left(\frac{a}{2} + \frac{\pi}{2} \right)}}{\sin^{4}{\left(a \right)}} + \frac{\sin{\left(2 a + \frac{\pi}{2} \right)}}{2} + \frac{1}{2}$$
4 8 8 1 cos(2*a) / 2/a\\ / 2/a\\ 16/a\ / 2/a\\ 16/a\ 4/a\ - + -------- - |1 - tan |-|| *|1 - tan |-|| *cos |-| + 16*|1 - tan |-|| *cos |-|*tan |-| 2 2 \ \2// \ \4// \4/ \ \4// \4/ \2/
$$- \left(- \tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{8} \left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4} \cos^{16}{\left(\frac{a}{4} \right)} + 16 \left(- \tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{8} \cos^{16}{\left(\frac{a}{4} \right)} \tan^{4}{\left(\frac{a}{2} \right)} + \frac{\cos{\left(2 a \right)}}{2} + \frac{1}{2}$$
4/a pi\ 2/a pi\
16*tan |- + --| 4*tan |- + --|
\2 4 / \2 4 / 4/a\ 8/a\
- ------------------- + ------------------- + 16*cot |-|*sin |-|
4 2 \2/ \2/
/ 2/a pi\\ / 2/a pi\\
|1 + tan |- + --|| |1 + tan |- + --||
\ \2 4 // \ \2 4 //
$$16 \sin^{8}{\left(\frac{a}{2} \right)} \cot^{4}{\left(\frac{a}{2} \right)} + \frac{4 \tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} - \frac{16 \tan^{4}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{4}}$$
4/a pi\ 2/a pi\
16*tan |- + --| 4*tan |- + --|
\2 4 / \2 4 / 8/a\ 4/a\
- ------------------- + ------------------- + 16*cos |-|*tan |-|
4 2 \2/ \2/
/ 2/a pi\\ / 2/a pi\\
|1 + tan |- + --|| |1 + tan |- + --||
\ \2 4 // \ \2 4 //
$$16 \cos^{8}{\left(\frac{a}{2} \right)} \tan^{4}{\left(\frac{a}{2} \right)} + \frac{4 \tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} - \frac{16 \tan^{4}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{4}}$$
4
/ 2/pi a\\
| csc |-- - -||
| \2 2/|
|1 - ------------|
| 2/a\ |
| csc |-| |
1 1 \ \2/ / 16
- + --------------- - ------------------- + --------------------
2 /pi \ 8/pi a\ 4/a\ 4/pi a\
2*csc|-- - 2*a| csc |-- - -| csc |-|*csc |-- - -|
\2 / \2 2/ \2/ \2 2/
$$\frac{1}{2} + \frac{1}{2 \csc{\left(- 2 a + \frac{\pi}{2} \right)}} - \frac{\left(1 - \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right)^{4}}{\csc^{8}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + \frac{16}{\csc^{4}{\left(\frac{a}{2} \right)} \csc^{4}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}$$
2 4
/ 2/a\\ / 2/a\\ 4/a\
|1 - tan |-|| |1 - tan |-|| 16*tan |-|
\ \2// \ \2// \2/
-------------- - -------------- + --------------
2 4 4
/ 2/a\\ / 2/a\\ / 2/a\\
|1 + tan |-|| |1 + tan |-|| |1 + tan |-||
\ \2// \ \2// \ \2//
$$- \frac{\left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4}} + \frac{\left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} + \frac{16 \tan^{4}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4}}$$
4/a pi\ 2/a pi\ 4/a\
16*tan |- + --| 4*tan |- + --| 16*cot |-|
\2 4 / \2 4 / \2/
- ------------------- + ------------------- + --------------
4 2 4
/ 2/a pi\\ / 2/a pi\\ / 2/a\\
|1 + tan |- + --|| |1 + tan |- + --|| |1 + cot |-||
\ \2 4 // \ \2 4 // \ \2//
$$\frac{16 \cot^{4}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4}} + \frac{4 \tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} - \frac{16 \tan^{4}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{4}}$$
4/a pi\ 2/a pi\ 4/a\
16*tan |- + --| 4*tan |- + --| 16*tan |-|
\2 4 / \2 4 / \2/
- ------------------- + ------------------- + --------------
4 2 4
/ 2/a pi\\ / 2/a pi\\ / 2/a\\
|1 + tan |- + --|| |1 + tan |- + --|| |1 + tan |-||
\ \2 4 // \ \2 4 // \ \2//
$$\frac{4 \tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} - \frac{16 \tan^{4}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{4}} + \frac{16 \tan^{4}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4}}$$
2 4 4
/ 2/a\\ / 2/a pi\\ / 2/a\\
|-1 + cot |-|| |-1 + tan |- + --|| |-1 + cot |-||
\ \2// \ \2 4 // \ \2//
--------------- + -------------------- - ---------------
2 4 4
/ 2/a\\ / 2/a pi\\ / 2/a\\
|1 + cot |-|| |1 + tan |- + --|| |1 + cot |-||
\ \2// \ \2 4 // \ \2//
$$\frac{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1\right)^{4}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{4}} - \frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{4}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4}} + \frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}$$
2 4
/ 1 \ / 1 \
|1 - -------| |1 - -------|
| 2/a\| | 2/a\|
| cot |-|| | cot |-||
\ \2// \ \2// 16
-------------- - -------------- + ----------------------
2 4 4
/ 1 \ / 1 \ / 1 \ 4/a\
|1 + -------| |1 + -------| |1 + -------| *cot |-|
| 2/a\| | 2/a\| | 2/a\| \2/
| cot |-|| | cot |-|| | cot |-||
\ \2// \ \2// \ \2//
$$- \frac{\left(1 - \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right)^{4}}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right)^{4}} + \frac{\left(1 - \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right)^{2}} + \frac{16}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right)^{4} \cot^{4}{\left(\frac{a}{2} \right)}}$$
4 2 4
/ 2/a pi\\ / 2/a\\ / 2/a\\
|1 - cot |- + --|| |1 - tan |-|| |1 - tan |-||
\ \2 4 // \ \2// \ \2//
------------------- + -------------- - --------------
4 2 4
/ 2/a pi\\ / 2/a\\ / 2/a\\
|1 + cot |- + --|| |1 + tan |-|| |1 + tan |-||
\ \2 4 // \ \2// \ \2//
$$- \frac{\left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4}} + \frac{\left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} + \frac{\left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{4}}{\left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{4}}$$
// 0 for a mod pi = 0\
|| |
// 1 for a mod 2*pi = 0\ || 4 | // 1 for a mod 2*pi = 0\
|| | ||(-1 + cos(a)) | || |
- |< 4 | + |<-------------- otherwise | + |< 2 |
||cos (a) otherwise | || 4/a\ | ||cos (a) otherwise |
\\ / || tan |-| | \\ /
|| \2/ |
\\ /
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{\left(\cos{\left(a \right)} - 1\right)^{4}}{\tan^{4}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos^{4}{\left(a \right)} & \text{otherwise} \end{cases}\right)$$
4 8 8
/ 2/a\\ / 2/a\\ / 2/a\\ 4/a\
2 |1 - tan |-|| *|1 - tan |-|| 16*|1 - tan |-|| *tan |-|
1 1 - tan (a) \ \2// \ \4// \ \4// \2/
- + --------------- - ----------------------------- + -------------------------
2 / 2 \ 8 8
2*\1 + tan (a)/ / 2/a\\ / 2/a\\
|1 + tan |-|| |1 + tan |-||
\ \4// \ \4//
$$- \frac{\left(- \tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{8} \left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4}}{\left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{8}} + \frac{16 \left(- \tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{8} \tan^{4}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{8}} + \frac{- \tan^{2}{\left(a \right)} + 1}{2 \left(\tan^{2}{\left(a \right)} + 1\right)} + \frac{1}{2}$$
// 1 for a mod 2*pi = 0\ || | || 4 | // 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\ ||/ 2 \ | || | || | - |<| sin (a) | 8/a\ | + |< 4 | + |< 2 | |||-1 + ---------| *sin |-| otherwise | ||sin (a) otherwise | ||cos (a) otherwise | ||| 4/a\| \2/ | \\ / \\ / ||| 4*sin |-|| | \\\ \2// /
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin^{4}{\left(a \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\left(-1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right)^{4} \sin^{8}{\left(\frac{a}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\
|| | // 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\
|| 4 | || | || |
- |
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\16 \sin^{8}{\left(\frac{a}{2} \right)} \cot^{4}{\left(\frac{a}{2} \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{4} \sin^{8}{\left(\frac{a}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\
|| | // 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\
|| 4 | || | || |
- |
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\16 \sin^{4}{\left(\frac{a}{2} \right)} \cos^{4}{\left(\frac{a}{2} \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4} \cos^{8}{\left(\frac{a}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\ || | || 4 | // 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\ ||/ 2 \ | || | || | - |<| sin (a) | 8/a\ | + |< 4 | + |< 2/ pi\ | |||-1 + ---------| *sin |-| otherwise | ||sin (a) otherwise | ||sin |a + --| otherwise | ||| 4/a\| \2/ | \\ / \\ \ 2 / / ||| 4*sin |-|| | \\\ \2// /
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin^{4}{\left(a \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\left(-1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right)^{4} \sin^{8}{\left(\frac{a}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\sin^{2}{\left(a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\
// 1 for a mod 2*pi = 0\ || |
|| | || 8/a\ | // 1 for a mod 2*pi = 0\
|| 4 | ||16*sin |-| | || |
- |
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{16 \sin^{8}{\left(\frac{a}{2} \right)}}{\tan^{4}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4} \cos^{8}{\left(\frac{a}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\ || | || 4 8/a\ | // 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\ ||16*cos (a)*sin |-| | || | || | - |< \2/ | + |< 4/a\ 4/a\ | + |< 2 | ||------------------ otherwise | ||16*cos |-|*sin |-| otherwise | ||cos (a) otherwise | || 4 | \\ \2/ \2/ / \\ / || (-1 + cos(a)) | \\ /
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\16 \sin^{4}{\left(\frac{a}{2} \right)} \cos^{4}{\left(\frac{a}{2} \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{16 \sin^{8}{\left(\frac{a}{2} \right)} \cos^{4}{\left(a \right)}}{\left(\cos{\left(a \right)} - 1\right)^{4}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right)$$
// / 3*pi\ \
|| 1 for |a + ----| mod 2*pi = 0|
|| \ 2 / |
// 1 for a mod 2*pi = 0\ || |
|| | // 1 for a mod 2*pi = 0\ || 4/a\ |
|| 4 | || | || 16*tan |-| |
- |
$$\left(- \begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{4} \sin^{8}{\left(\frac{a}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{16 \tan^{4}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4}} & \text{otherwise} \end{cases}\right)$$
2 4
/ 4/a\\ / 4/a\\
| 4*sin |-|| | 4*sin |-||
| \2/| | \2/|
|1 - ---------| |1 - ---------| 8/a\
| 2 | | 2 | 256*sin |-|
\ sin (a) / \ sin (a) / \2/
---------------- - ---------------- + ------------------------
2 4 4
/ 4/a\\ / 4/a\\ / 4/a\\
| 4*sin |-|| | 4*sin |-|| | 4*sin |-||
| \2/| | \2/| | \2/| 4
|1 + ---------| |1 + ---------| |1 + ---------| *sin (a)
| 2 | | 2 | | 2 |
\ sin (a) / \ sin (a) / \ sin (a) /
$$- \frac{\left(- \frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right)^{4}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right)^{4}} + \frac{\left(- \frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right)^{2}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right)^{2}} + \frac{256 \sin^{8}{\left(\frac{a}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right)^{4} \sin^{4}{\left(a \right)}}$$
// 1 for a mod 2*pi = 0\ || | || 4 | ||/ 2/a\ \ | // 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\ ||| cos |-| | | || | || | - |<| \2/ | 8/a pi\ | + |< 4/a\ 4/a pi\ | + |< 2 | |||-1 + ------------| *cos |- - --| otherwise | ||16*cos |-|*cos |- - --| otherwise | ||cos (a) otherwise | ||| 2/a pi\| \2 2 / | \\ \2/ \2 2 / / \\ / ||| cos |- - --|| | ||\ \2 2 // | \\ /
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\16 \cos^{4}{\left(\frac{a}{2} \right)} \cos^{4}{\left(\frac{a}{2} - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - 1\right)^{4} \cos^{8}{\left(\frac{a}{2} - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\ // 0 for a mod pi = 0\ || | || | || 4 | || 16/a\ 8/a\ | // 1 for a mod 2*pi = 0\ || / 1 \ 16/a\ 8/a\ | ||4096*cos |-|*tan |-| | || | - |<256*|-1 + -------| *cos |-|*tan |-| otherwise | + |< \4/ \4/ | + |< 2 | || | 2/a\| \4/ \4/ | ||--------------------- otherwise | ||cos (a) otherwise | || | tan |-|| | || 4/a\ | \\ / || \ \2// | || tan |-| | \\ / \\ \2/ /
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{4096 \cos^{16}{\left(\frac{a}{4} \right)} \tan^{8}{\left(\frac{a}{4} \right)}}{\tan^{4}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\256 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right)^{4} \cos^{16}{\left(\frac{a}{4} \right)} \tan^{8}{\left(\frac{a}{4} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\ || | || 4 | ||/ 2/a pi\\ | ||| sec |- - --|| | // 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\ ||| \2 2 /| | || | || | |||-1 + ------------| | || 16 | || 1 | - |<| 2/a\ | | + |<-------------------- otherwise | + |<------- otherwise | ||| sec |-| | | || 4/a\ 4/a pi\ | || 2 | ||\ \2/ / | ||sec |-|*sec |- - --| | ||sec (a) | ||-------------------- otherwise | \\ \2/ \2 2 / / \\ / || 8/a pi\ | || sec |- - --| | || \2 2 / | \\ /
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{16}{\sec^{4}{\left(\frac{a}{2} \right)} \sec^{4}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right)^{4}}{\sec^{8}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\sec^{2}{\left(a \right)}} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\ || | || 4 | ||/ 2/a\ \ | ||| csc |-| | | // 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\ ||| \2/ | | || | || | |||-1 + ------------| | || 16 | || 1 | - |<| 2/pi a\| | + |<-------------------- otherwise | + |<------------ otherwise | ||| csc |-- - -|| | || 4/a\ 4/pi a\ | || 2/pi \ | ||\ \2 2// | ||csc |-|*csc |-- - -| | ||csc |-- - a| | ||-------------------- otherwise | \\ \2/ \2 2/ / \\ \2 / / || 8/a\ | || csc |-| | || \2/ | \\ /
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{16}{\csc^{4}{\left(\frac{a}{2} \right)} \csc^{4}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} - 1\right)^{4}}{\csc^{8}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// / pi\ \ // / pi\ \ || 0 for |a + --| mod pi = 0| // 0 for a mod pi = 0\ || 0 for |a + --| mod pi = 0| || \ 2 / | || | || \ 2 / | - |< | + |< 4/a\ 8/a\ | + |< | || 4 4/a pi\ | ||16*cot |-|*sin |-| otherwise | || 2 2/a pi\ | ||(1 + sin(a)) *cot |- + --| otherwise | \\ \2/ \2/ / ||(1 + sin(a)) *cot |- + --| otherwise | \\ \2 4 / / \\ \2 4 / /
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\16 \sin^{8}{\left(\frac{a}{2} \right)} \cot^{4}{\left(\frac{a}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(a \right)} + 1\right)^{2} \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(a \right)} + 1\right)^{4} \cot^{4}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\ // 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\ || | || | || | || 4 | || 4/a\ | || 2 | ||/ 2/a\\ | || 16*cot |-| | ||/ 2/a\\ | |||-1 + cot |-|| | || \2/ | |||-1 + cot |-|| | - |<\ \2// | + |<-------------- otherwise | + |<\ \2// | ||--------------- otherwise | || 4 | ||--------------- otherwise | || 4 | ||/ 2/a\\ | || 2 | || / 2/a\\ | |||1 + cot |-|| | || / 2/a\\ | || |1 + cot |-|| | ||\ \2// | || |1 + cot |-|| | \\ \ \2// / \\ / \\ \ \2// /
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{16 \cot^{4}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{4}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4}} & \text{otherwise} \end{cases}\right)$$
// a \ // a \
/ 1 for a mod pi = 0 || 1 for - mod 2*pi = 0| || 1 for - mod 2*pi = 0|
< 4 || 2 | || 2 |
1 \cos(2*a) otherwise / 2/a\\ || | 4/a\ || |
- + --------------------------- - |1 - tan |-|| *|< 8 | + 16*tan |-|*|< 8 |
2 2 \ \2// ||/ 2/a\\ 16/a\ | \2/ ||/ 2/a\\ 16/a\ |
|||-1 + cot |-|| *sin |-| otherwise | |||-1 + cot |-|| *sin |-| otherwise |
||\ \4// \4/ | ||\ \4// \4/ |
\\ / \\ /
$$\left(- \left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4} \left(\begin{cases} 1 & \text{for}\: \frac{a}{2} \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{8} \sin^{16}{\left(\frac{a}{4} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(16 \left(\begin{cases} 1 & \text{for}\: \frac{a}{2} \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{8} \sin^{16}{\left(\frac{a}{4} \right)} & \text{otherwise} \end{cases}\right) \tan^{4}{\left(\frac{a}{2} \right)}\right) + \left(\frac{\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}}{2}\right) + \frac{1}{2}$$
// 1 for a mod 2*pi = 0\ // 1 for a mod 2*pi = 0\
|| | || |
|| 4 | // 0 for a mod pi = 0\ || 2 |
||/ 1 \ | || | ||/ 1 \ |
|||-1 + -------| | || 16 | |||-1 + -------| |
||| 2/a\| | ||---------------------- otherwise | ||| 2/a\| |
||| tan |-|| | || 4 | ||| tan |-|| |
- |<\ \2// | + |
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{16}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right)^{4} \tan^{4}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right)^{4}}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right)^{4}} & \text{otherwise} \end{cases}\right)$$
2 4
/ 2/a pi\\ / 2/a pi\\
| cos |- - --|| | cos |- - --||
| \2 2 /| | \2 2 /|
|1 - ------------| |1 - ------------|
| 2/a\ | | 2/a\ | 4/a pi\
| cos |-| | | cos |-| | 16*cos |- - --|
\ \2/ / \ \2/ / \2 2 /
------------------- - ------------------- + ---------------------------
2 4 4
/ 2/a pi\\ / 2/a pi\\ / 2/a pi\\
| cos |- - --|| | cos |- - --|| | cos |- - --||
| \2 2 /| | \2 2 /| | \2 2 /| 4/a\
|1 + ------------| |1 + ------------| |1 + ------------| *cos |-|
| 2/a\ | | 2/a\ | | 2/a\ | \2/
| cos |-| | | cos |-| | | cos |-| |
\ \2/ / \ \2/ / \ \2/ /
$$- \frac{\left(1 - \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right)^{4}}{\left(1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right)^{4}} + \frac{\left(1 - \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right)^{2}}{\left(1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right)^{2}} + \frac{16 \cos^{4}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right)^{4} \cos^{4}{\left(\frac{a}{2} \right)}}$$
2 4
/ 2/a\ \ / 2/a\ \
| sec |-| | | sec |-| |
| \2/ | | \2/ |
|1 - ------------| |1 - ------------|
| 2/a pi\| | 2/a pi\| 4/a\
| sec |- - --|| | sec |- - --|| 16*sec |-|
\ \2 2 // \ \2 2 // \2/
------------------- - ------------------- + --------------------------------
2 4 4
/ 2/a\ \ / 2/a\ \ / 2/a\ \
| sec |-| | | sec |-| | | sec |-| |
| \2/ | | \2/ | | \2/ | 4/a pi\
|1 + ------------| |1 + ------------| |1 + ------------| *sec |- - --|
| 2/a pi\| | 2/a pi\| | 2/a pi\| \2 2 /
| sec |- - --|| | sec |- - --|| | sec |- - --||
\ \2 2 // \ \2 2 // \ \2 2 //
$$- \frac{\left(- \frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)^{4}}{\left(\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)^{4}} + \frac{\left(- \frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}}{\left(\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}} + \frac{16 \sec^{4}{\left(\frac{a}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)^{4} \sec^{4}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}$$
// / pi\ \ // / pi\ \ || 0 for |a + --| mod pi = 0| // 0 for a mod pi = 0\ || 0 for |a + --| mod pi = 0| || \ 2 / | || | || \ 2 / | || | || 4/a\ | || | || 4/a pi\ | || 16*cot |-| | || 2/a pi\ | || 16*cot |- + --| | || \2/ | || 4*cot |- + --| | - |< \2 4 / | + |<-------------- otherwise | + |< \2 4 / | ||------------------- otherwise | || 4 | ||------------------- otherwise | || 4 | ||/ 2/a\\ | || 2 | ||/ 2/a pi\\ | |||1 + cot |-|| | ||/ 2/a pi\\ | |||1 + cot |- + --|| | ||\ \2// | |||1 + cot |- + --|| | ||\ \2 4 // | \\ / ||\ \2 4 // | \\ / \\ /
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{16 \cot^{4}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{16 \cot^{4}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{4}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// / 3*pi\ \
// 1 for a mod 2*pi = 0\ // 1 for a mod 2*pi = 0\ || 1 for |a + ----| mod 2*pi = 0|
|| | || | || \ 2 / |
|| 4 | || 2 | || |
||/ 2/a\\ | ||/ 2/a\\ | || 4 |
|||-1 + cot |-|| | |||-1 + cot |-|| | ||/ 2/a pi\\ |
- |<\ \2// | + |<\ \2// | + |<|-1 + tan |- + --|| |
||--------------- otherwise | ||--------------- otherwise | ||\ \2 4 // |
|| 4 | || 2 | ||-------------------- otherwise |
|| / 2/a\\ | || / 2/a\\ | || 4 |
|| |1 + cot |-|| | || |1 + cot |-|| | ||/ 2/a pi\\ |
\\ \ \2// / \\ \ \2// / |||1 + tan |- + --|| |
\\\ \2 4 // /
$$\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{4}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1\right)^{4}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{4}} & \text{otherwise} \end{cases}\right)$$
2 4
/ 2/pi a\\ / 2/pi a\\
| csc |-- - -|| | csc |-- - -||
| \2 2/| | \2 2/|
|1 - ------------| |1 - ------------|
| 2/a\ | | 2/a\ | 4/pi a\
| csc |-| | | csc |-| | 16*csc |-- - -|
\ \2/ / \ \2/ / \2 2/
------------------- - ------------------- + ---------------------------
2 4 4
/ 2/pi a\\ / 2/pi a\\ / 2/pi a\\
| csc |-- - -|| | csc |-- - -|| | csc |-- - -||
| \2 2/| | \2 2/| | \2 2/| 4/a\
|1 + ------------| |1 + ------------| |1 + ------------| *csc |-|
| 2/a\ | | 2/a\ | | 2/a\ | \2/
| csc |-| | | csc |-| | | csc |-| |
\ \2/ / \ \2/ / \ \2/ /
$$- \frac{\left(1 - \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right)^{4}}{\left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right)^{4}} + \frac{\left(1 - \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right)^{2}}{\left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right)^{2}} + \frac{16 \csc^{4}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right)^{4} \csc^{4}{\left(\frac{a}{2} \right)}}$$
// 1 for a mod 2*pi = 0\ || | // 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\ || 4 | || | || | || / 1 \ 8/a\ | || 8/a\ | || 2 | ||256*|-1 + -------| *tan |-| | || 4096*tan |-| | ||/ 2/a\\ | || | 2/a\| \4/ | || \4/ | |||1 - tan |-|| | - |< | tan |-|| | + |<---------------------- otherwise | + |<\ \2// | || \ \2// | || 8 | ||-------------- otherwise | ||--------------------------- otherwise | ||/ 2/a\\ 4/a\ | || 2 | || 8 | |||1 + tan |-|| *tan |-| | ||/ 2/a\\ | || / 2/a\\ | ||\ \4// \2/ | |||1 + tan |-|| | || |1 + tan |-|| | \\ / \\\ \2// / \\ \ \4// /
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{4096 \tan^{8}{\left(\frac{a}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{8} \tan^{4}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{256 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right)^{4} \tan^{8}{\left(\frac{a}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{8}} & \text{otherwise} \end{cases}\right)$$
// a \
|| 1 for - mod 2*pi = 0|
|| 2 |
|| |
|| 8 |
||/ 2/a\\ |
/ 1 for a mod pi = 0 16*|<|-1 + cot |-|| |
| // a \ ||\ \4// |
| 2 || 1 for - mod 2*pi = 0| ||--------------- otherwise |
<-1 + cot (a) || 2 | || 8 |
|------------ otherwise || | || / 2/a\\ |
| 2 4 || 8 | || |1 + cot |-|| |
1 \1 + cot (a) / 1 \ ||/ 2/a\\ | \\ \ \4// /
- + ------------------------------- - |1 - -------| *|<|-1 + cot |-|| | + -----------------------------------------
2 2 | 2/a\| ||\ \4// | 4/a\
| cot |-|| ||--------------- otherwise | cot |-|
\ \2// || 8 | \2/
|| / 2/a\\ |
|| |1 + cot |-|| |
\\ \ \4// /
$$\left(- \left(1 - \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right)^{4} \left(\begin{cases} 1 & \text{for}\: \frac{a}{2} \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{8}}{\left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{8}} & \text{otherwise} \end{cases}\right)\right) + \left(\frac{\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}}{2}\right) + \frac{1}{2} + \left(\frac{16 \left(\begin{cases} 1 & \text{for}\: \frac{a}{2} \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{8}}{\left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{8}} & \text{otherwise} \end{cases}\right)}{\cot^{4}{\left(\frac{a}{2} \right)}}\right)$$
// 0 for a mod pi = 0\
// 1 for a mod 2*pi = 0\ || | // 1 for a mod 2*pi = 0\
|| | || 4 | || |
|| 4 | || sin (a) | || 2 |
||/ 2 4/a\\ | ||------------------------ otherwise | ||/ 2 4/a\\ |
|||sin (a) - 4*sin |-|| | || 4 | |||sin (a) - 4*sin |-|| |
- |<\ \2// | + |
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{\sin^{4}{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right)^{4} \sin^{8}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(- 4 \sin^{4}{\left(\frac{a}{2} \right)} + \sin^{2}{\left(a \right)}\right)^{2}}{\left(4 \sin^{4}{\left(\frac{a}{2} \right)} + \sin^{2}{\left(a \right)}\right)^{2}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(- 4 \sin^{4}{\left(\frac{a}{2} \right)} + \sin^{2}{\left(a \right)}\right)^{4}}{\left(4 \sin^{4}{\left(\frac{a}{2} \right)} + \sin^{2}{\left(a \right)}\right)^{4}} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\ // 1 for a mod 2*pi = 0\
|| | || |
|| 4 | // 0 for a mod pi = 0\ || 2 |
||/ 2 \ | || | ||/ 2 \ |
||| sin (a) | | || 4 | ||| sin (a) | |
|||-1 + ---------| | || sin (a) | |||-1 + ---------| |
||| 4/a\| | ||------------------------ otherwise | ||| 4/a\| |
||| 4*sin |-|| | || 4 | ||| 4*sin |-|| |
- |<\ \2// | + |
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{\sin^{4}{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right)^{4} \sin^{8}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right)^{4}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right)^{4}} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\ // 0 for a mod pi = 0\
|| | || | // 1 for a mod 2*pi = 0\
|| // a \ | || // a \ | || |
|| 4 || 0 for - mod pi = 0| | || || 0 for - mod pi = 0| | ||/ 1 for a mod 2*pi = 0 |
- |
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\16 \left(\begin{cases} 0 & \text{for}\: \frac{a}{2} \bmod \pi = 0 \\256 \sin^{16}{\left(\frac{a}{4} \right)} \cot^{8}{\left(\frac{a}{4} \right)} & \text{otherwise} \end{cases}\right) \cot^{4}{\left(\frac{a}{2} \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{4} \left(\begin{cases} 0 & \text{for}\: \frac{a}{2} \bmod \pi = 0 \\256 \sin^{16}{\left(\frac{a}{4} \right)} \cot^{8}{\left(\frac{a}{4} \right)} & \text{otherwise} \end{cases}\right) & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos^{2}{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\ // 1 for a mod 2*pi = 0\
|| | || |
|| 4 | // 0 for a mod pi = 0\ || 2 |
||/ 2/a\ \ | || | ||/ 2/a\ \ |
||| cos |-| | | || 4/a\ | ||| cos |-| | |
||| \2/ | | || 16*cos |-| | ||| \2/ | |
|||-1 + ------------| | || \2/ | |||-1 + ------------| |
||| 2/a pi\| | ||-------------------------------- otherwise | ||| 2/a pi\| |
||| cos |- - --|| | || 4 | ||| cos |- - --|| |
- |<\ \2 2 // | + |
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{16 \cos^{4}{\left(\frac{a}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)^{4} \cos^{4}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - 1\right)^{4}}{\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)^{4}} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\ // 1 for a mod 2*pi = 0\
|| | || |
|| 4 | // 0 for a mod pi = 0\ || 2 |
||/ 2/a pi\\ | || | ||/ 2/a pi\\ |
||| sec |- - --|| | || 4/a pi\ | ||| sec |- - --|| |
||| \2 2 /| | || 16*sec |- - --| | ||| \2 2 /| |
|||-1 + ------------| | || \2 2 / | |||-1 + ------------| |
||| 2/a\ | | ||--------------------------- otherwise | ||| 2/a\ | |
||| sec |-| | | || 4 | ||| sec |-| | |
- |<\ \2/ / | + |
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{16 \sec^{4}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right)^{4} \sec^{4}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right)^{4}}{\left(1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right)^{4}} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\ // 1 for a mod 2*pi = 0\
|| | || |
|| 4 | // 0 for a mod pi = 0\ || 2 |
||/ 2/a\ \ | || | ||/ 2/a\ \ |
||| csc |-| | | || 4/a\ | ||| csc |-| | |
||| \2/ | | || 16*csc |-| | ||| \2/ | |
|||-1 + ------------| | || \2/ | |||-1 + ------------| |
||| 2/pi a\| | ||-------------------------------- otherwise | ||| 2/pi a\| |
||| csc |-- - -|| | || 4 | ||| csc |-- - -|| |
- |<\ \2 2// | + |
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{16 \csc^{4}{\left(\frac{a}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1\right)^{4} \csc^{4}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} - 1\right)^{4}}{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1\right)^{4}} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\ // 0 for a mod pi = 0\
|| | || | // 1 for a mod 2*pi = 0\
|| // a \ | || // a \ | || |
|| || 0 for - mod pi = 0| | || || 0 for - mod pi = 0| | ||/ 1 for a mod 2*pi = 0 |
|| || 2 | | || || 2 | | ||| |
|| || | | || || | | ||| 2 |
|| 4 || 8/a\ | | || || 8/a\ | | |||/ 2/a\\ |
- |
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\16 \left(\begin{cases} 0 & \text{for}\: \frac{a}{2} \bmod \pi = 0 \\\frac{256 \cot^{8}{\left(\frac{a}{4} \right)}}{\left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{8}} & \text{otherwise} \end{cases}\right) \cot^{4}{\left(\frac{a}{2} \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{4} \left(\begin{cases} 0 & \text{for}\: \frac{a}{2} \bmod \pi = 0 \\\frac{256 \cot^{8}{\left(\frac{a}{4} \right)}}{\left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{8}} & \text{otherwise} \end{cases}\right) & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
-Piecewise((1, Mod(a = 2*pi, 0)), ((-1 + cot(a/2)^2)^4*Piecewise((0, Mod(a/2 = pi, 0)), (256*cot(a/4)^8/(1 + cot(a/4)^2)^8, True)), True)) + Piecewise((0, Mod(a = pi, 0)), (16*cot(a/2)^4*Piecewise((0, Mod(a/2 = pi, 0)), (256*cot(a/4)^8/(1 + cot(a/4)^2)^8, True)), True)) + Piecewise((1, Mod(a = 2*pi, 0)), (Piecewise((1, Mod(a = 2*pi, 0)), ((-1 + cot(a/2)^2)^2/(1 + cot(a/2)^2)^2, True)), True))