Господин Экзамен
Другие калькуляторы
-
Как пользоваться?
- sin(a)^4+cos(a)^4 если a=1/2
- 7*(d+7)^2-7*d^2 если d=3
- 36*b*2 если b=2
- sin(t)^2-1/(cos(t)^2)-1+tan(t)*cot(t) если t=1/4
- Разложить многочлен на множители:
- x^2-5*x+6
- 2*x^2+11*x+15
- 9-b^4
- a^3-8
- Общий знаменатель:
- 1-6/x/12*x-36/x-x
- cos(p)/7*cos(p)/42-sin(p)/7*sin(p)/42
- x^2-16/x^3-3*x^2*x^2-9/x^2+4*x
- 9*a^2+y^2/3*a-y+6*a*y/y-3*a
- Умножение столбиком:
- 15*30
- 50*10
- 25*32
- 30*75
- Деление столбиком:
- 12012/3
- 835/4
- 63/2
- 21/3
- Раскрыть скобки в:
- z*(-z)*(-z)^8
- (b+3)*(b-3)
- 6*(c+1)-6*c-5
- 7*(2*a+5)^2+5*(2*a-7)^2
- Разложение числа на простые множители:
- 4356
- 2340
- 2520
- 2280
- График функции y =:
- 1/2
- Производная:
- 1/2
- Интеграл d{x}:
-
1/2
-
Идентичные выражения
- sin(a)^ четыре +cos(a)^ четыре если a= один / два
- синус от (a) в степени 4 плюс косинус от (a) в степени 4 если a равно 1 делить на 2
- синус от (a) в степени четыре плюс косинус от (a) в степени четыре если a равно один делить на два
- sin(a)4+cos(a)4 если a=1/2
- sina4+cosa4 если a=1/2
- sin(a)⁴+cos(a)⁴ если a=1/2
- sina^4+cosa^4 если a=1/2
- sin(a)^4+cos(a)^4 при a=1/2
- sin(a)^4+cos(a)^4 если a=1 разделить на 2
-
Похожие выражения
- sin(a)^4-cos(a)^4 если a=1/2
- sqrt(sin(a)^2-sin(a)^4+cos(a)^4) если a=-4
sin(a)^4+cos(a)^4 если a=1/2
Решение
Вы ввели
[src]
4 4 sin (a) + cos (a)
$$\sin^{4}{\left(a \right)} + \cos^{4}{\left(a \right)}$$
sin(a)^4 + cos(a)^4
Подстановка условия
[src]
sin(a)^4 + cos(a)^4 при a = 1/2
подставляем
4 4 sin (a) + cos (a)
$$\sin^{4}{\left(a \right)} + \cos^{4}{\left(a \right)}$$
4 4 cos (a) + sin (a)
$$\sin^{4}{\left(a \right)} + \cos^{4}{\left(a \right)}$$
переменные
a = 1/2
$$a = \frac{1}{2}$$
4 4 cos ((1/2)) + sin ((1/2))
$$\sin^{4}{\left((1/2) \right)} + \cos^{4}{\left((1/2) \right)}$$
4 4 cos (1/2) + sin (1/2)
$$\sin^{4}{\left(\frac{1}{2} \right)} + \cos^{4}{\left(\frac{1}{2} \right)}$$
cos(1/2)^4 + sin(1/2)^4
Собрать выражение
[src]
3 cos(4*a) - + -------- 4 4
$$\frac{\cos{\left(4 a \right)}}{4} + \frac{3}{4}$$
3/4 + cos(4*a)/4
Степени
[src]
4 4 / I*a -I*a\ / -I*a I*a\ |e e | \- e + e / |---- + -----| + ----------------- \ 2 2 / 16
$$\left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2}\right)^{4} + \frac{\left(e^{i a} - e^{- i a}\right)^{4}}{16}$$
(exp(i*a)/2 + exp(-i*a)/2)^4 + (-exp(-i*a) + exp(i*a))^4/16
Тригонометрическая часть
[src]
3 cos(4*a) - + -------- 4 4
$$\frac{\cos{\left(4 a \right)}}{4} + \frac{3}{4}$$
4 4/ pi\
sin (a) + sin |a + --|
\ 2 /
$$\sin^{4}{\left(a \right)} + \sin^{4}{\left(a + \frac{\pi}{2} \right)}$$
4 4/ pi\
cos (a) + cos |a - --|
\ 2 /
$$\cos^{4}{\left(a \right)} + \cos^{4}{\left(a - \frac{\pi}{2} \right)}$$
1 1 ------- + ------- 4 4 csc (a) sec (a)
$$\frac{1}{\sec^{4}{\left(a \right)}} + \frac{1}{\csc^{4}{\left(a \right)}}$$
1 1
------- + ------------
4 4/ pi\
sec (a) sec |a - --|
\ 2 /
$$\frac{1}{\sec^{4}{\left(a - \frac{\pi}{2} \right)}} + \frac{1}{\sec^{4}{\left(a \right)}}$$
1 1
------- + ------------
4 4/pi \
csc (a) csc |-- - a|
\2 /
$$\frac{1}{\csc^{4}{\left(- a + \frac{\pi}{2} \right)}} + \frac{1}{\csc^{4}{\left(a \right)}}$$
1 1
------- + ------------
4 4/pi \
sec (a) sec |-- - a|
\2 /
$$\frac{1}{\sec^{4}{\left(- a + \frac{\pi}{2} \right)}} + \frac{1}{\sec^{4}{\left(a \right)}}$$
1 1
------------ + ------------
4 4/pi \
csc (pi - a) csc |-- - a|
\2 /
$$\frac{1}{\csc^{4}{\left(- a + \frac{\pi}{2} \right)}} + \frac{1}{\csc^{4}{\left(- a + \pi \right)}}$$
4 4 4/a\
cos (a) + (1 + cos(a)) *tan |-|
\2/
$$\left(\cos{\left(a \right)} + 1\right)^{4} \tan^{4}{\left(\frac{a}{2} \right)} + \cos^{4}{\left(a \right)}$$
2
/ 2 2 \
3 \cos (a) - sin (a)/ 2 2
- + -------------------- - cos (a)*sin (a)
4 4
$$- \sin^{2}{\left(a \right)} \cos^{2}{\left(a \right)} + \frac{\left(- \sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}\right)^{2}}{4} + \frac{3}{4}$$
2 2 4/a\ 4/a\
(1 - sin(a)) *(1 + sin(a)) + 16*cos |-|*sin |-|
\2/ \2/
$$16 \sin^{4}{\left(\frac{a}{2} \right)} \cos^{4}{\left(\frac{a}{2} \right)} + \left(- \sin{\left(a \right)} + 1\right)^{2} \left(\sin{\left(a \right)} + 1\right)^{2}$$
4 / 2/a\\ 8/a\ 4/a\ 4/a\ |1 - tan |-|| *cos |-| + 16*cos |-|*sin |-| \ \2// \2/ \2/ \2/
$$\left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4} \cos^{8}{\left(\frac{a}{2} \right)} + 16 \sin^{4}{\left(\frac{a}{2} \right)} \cos^{4}{\left(\frac{a}{2} \right)}$$
4 / 2/a\\ 8/a\ 8/a\ 4/a\ |1 - tan |-|| *cos |-| + 16*cos |-|*tan |-| \ \2// \2/ \2/ \2/
$$\left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4} \cos^{8}{\left(\frac{a}{2} \right)} + 16 \cos^{8}{\left(\frac{a}{2} \right)} \tan^{4}{\left(\frac{a}{2} \right)}$$
4 8/a\
16*cos (a)*cos |-|
4/a\ 4/a\ \2/
16*cos |-|*sin |-| + ------------------
\2/ \2/ 4
(1 + cos(a))
$$16 \sin^{4}{\left(\frac{a}{2} \right)} \cos^{4}{\left(\frac{a}{2} \right)} + \frac{16 \cos^{8}{\left(\frac{a}{2} \right)} \cos^{4}{\left(a \right)}}{\left(\cos{\left(a \right)} + 1\right)^{4}}$$
4/a\
4 16*tan |-|
/ 2/a\\ 8/a\ \2/
|-1 + cot |-|| *sin |-| + --------------
\ \2// \2/ 4
/ 2/a\\
|1 + tan |-||
\ \2//
$$\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{4} \sin^{8}{\left(\frac{a}{2} \right)} + \frac{16 \tan^{4}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4}}$$
4/a pi\
16*tan |- + --|
\2 4 / 4/a\ 8/a\
------------------- + 16*cot |-|*sin |-|
4 \2/ \2/
/ 2/a pi\\
|1 + tan |- + --||
\ \2 4 //
$$16 \sin^{8}{\left(\frac{a}{2} \right)} \cot^{4}{\left(\frac{a}{2} \right)} + \frac{16 \tan^{4}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{4}}$$
4/a pi\
16*tan |- + --|
\2 4 / 8/a\ 4/a\
------------------- + 16*cos |-|*tan |-|
4 \2/ \2/
/ 2/a pi\\
|1 + tan |- + --||
\ \2 4 //
$$16 \cos^{8}{\left(\frac{a}{2} \right)} \tan^{4}{\left(\frac{a}{2} \right)} + \frac{16 \tan^{4}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{4}}$$
4
/ 2/a pi\\ 4
4 |1 - cot |- + --|| *(1 + sin(a))
/ 2/a\\ 8/a\ \ \2 4 //
|1 - tan |-|| *cos |-| + ---------------------------------
\ \2// \2/ 16
$$\left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4} \cos^{8}{\left(\frac{a}{2} \right)} + \frac{\left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{4} \left(\sin{\left(a \right)} + 1\right)^{4}}{16}$$
4
/ 2/a\\ 4/a\
|1 - tan |-|| 16*tan |-|
\ \2// \2/
-------------- + --------------
4 4
/ 2/a\\ / 2/a\\
|1 + tan |-|| |1 + tan |-||
\ \2// \ \2//
$$\frac{\left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4}} + \frac{16 \tan^{4}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4}}$$
4/a\ 4/a pi\
16*tan |-| 16*tan |- + --|
\2/ \2 4 /
-------------- + -------------------
4 4
/ 2/a\\ / 2/a pi\\
|1 + tan |-|| |1 + tan |- + --||
\ \2// \ \2 4 //
$$\frac{16 \tan^{4}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{4}} + \frac{16 \tan^{4}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4}}$$
4/a\ 4/a pi\
16*cot |-| 16*tan |- + --|
\2/ \2 4 /
-------------- + -------------------
4 4
/ 2/a\\ / 2/a pi\\
|1 + cot |-|| |1 + tan |- + --||
\ \2// \ \2 4 //
$$\frac{16 \cot^{4}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4}} + \frac{16 \tan^{4}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{4}}$$
4 / 2/a pi\\ | cos |- - --|| | \2 2 /| 8/a\ 4/a\ 4/a pi\ |1 - ------------| *cos |-| + 16*cos |-|*cos |- - --| | 2/a\ | \2/ \2/ \2 2 / | cos |-| | \ \2/ /
$$\left(1 - \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right)^{4} \cos^{8}{\left(\frac{a}{2} \right)} + 16 \cos^{4}{\left(\frac{a}{2} \right)} \cos^{4}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}$$
4
/ 1 \
|1 - -------|
| 2/a\|
| cot |-||
\ \2// 16
-------------- + ----------------------
4 4
/ 1 \ / 1 \ 4/a\
|1 + -------| |1 + -------| *cot |-|
| 2/a\| | 2/a\| \2/
| cot |-|| | cot |-||
\ \2// \ \2//
$$\frac{\left(1 - \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right)^{4}}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right)^{4}} + \frac{16}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right)^{4} \cot^{4}{\left(\frac{a}{2} \right)}}$$
4
/ 2/a\ \
| sec |-| |
| \2/ |
|1 - ------------|
| 2/a pi\|
| sec |- - --||
\ \2 2 // 16
------------------- + --------------------
8/a\ 4/a\ 4/a pi\
sec |-| sec |-|*sec |- - --|
\2/ \2/ \2 2 /
$$\frac{\left(- \frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)^{4}}{\sec^{8}{\left(\frac{a}{2} \right)}} + \frac{16}{\sec^{4}{\left(\frac{a}{2} \right)} \sec^{4}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}$$
4 / 4/a\\ 8/a\ 8/pi a\ | 4*sin |-|| 256*sin |-|*sin |-- + -| | \2/| 8/pi a\ \2/ \2 2/ |1 - ---------| *sin |-- + -| + ------------------------ | 2 | \2 2/ 4 \ sin (a) / sin (a)
$$\left(- \frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right)^{4} \sin^{8}{\left(\frac{a}{2} + \frac{\pi}{2} \right)} + \frac{256 \sin^{8}{\left(\frac{a}{2} \right)} \sin^{8}{\left(\frac{a}{2} + \frac{\pi}{2} \right)}}{\sin^{4}{\left(a \right)}}$$
4 4
/ 2/a\\ / 2/a pi\\
|-1 + cot |-|| |-1 + tan |- + --||
\ \2// \ \2 4 //
--------------- + --------------------
4 4
/ 2/a\\ / 2/a pi\\
|1 + cot |-|| |1 + tan |- + --||
\ \2// \ \2 4 //
$$\frac{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1\right)^{4}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{4}} + \frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{4}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4}}$$
4 4
/ 2/a pi\\ / 2/a\\
|1 - cot |- + --|| |1 - tan |-||
\ \2 4 // \ \2//
------------------- + --------------
4 4
/ 2/a pi\\ / 2/a\\
|1 + cot |- + --|| |1 + tan |-||
\ \2 4 // \ \2//
$$\frac{\left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4}} + \frac{\left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{4}}{\left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{4}}$$
4
/ 2/pi a\\
| csc |-- - -||
| \2 2/|
|1 - ------------|
| 2/a\ |
| csc |-| |
\ \2/ / 16
------------------- + --------------------
8/pi a\ 4/a\ 4/pi a\
csc |-- - -| csc |-|*csc |-- - -|
\2 2/ \2/ \2 2/
$$\frac{\left(1 - \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right)^{4}}{\csc^{8}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + \frac{16}{\csc^{4}{\left(\frac{a}{2} \right)} \csc^{4}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}$$
4 8 8 / 2/a\\ / 2/a\\ 16/a\ / 2/a\\ 16/a\ 4/a\ |1 - tan |-|| *|1 - tan |-|| *cos |-| + 16*|1 - tan |-|| *cos |-|*tan |-| \ \2// \ \4// \4/ \ \4// \4/ \2/
$$\left(- \tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{8} \left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4} \cos^{16}{\left(\frac{a}{4} \right)} + 16 \left(- \tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{8} \cos^{16}{\left(\frac{a}{4} \right)} \tan^{4}{\left(\frac{a}{2} \right)}$$
// 0 for a mod pi = 0\ || | || 4 | // 1 for a mod 2*pi = 0\ ||(-1 + cos(a)) | || | |<-------------- otherwise | + |< 4 | || 4/a\ | ||cos (a) otherwise | || tan |-| | \\ / || \2/ | \\ /
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{\left(\cos{\left(a \right)} - 1\right)^{4}}{\tan^{4}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos^{4}{\left(a \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\
|| |
// 0 for a mod pi = 0\ || 4 |
|| | ||/ 2 \ |
|< 4 | + |<| sin (a) | 8/a\ |
||sin (a) otherwise | |||-1 + ---------| *sin |-| otherwise |
\\ / ||| 4/a\| \2/ |
||| 4*sin |-|| |
\\\ \2// /
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin^{4}{\left(a \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\left(-1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right)^{4} \sin^{8}{\left(\frac{a}{2} \right)} & \text{otherwise} \end{cases}\right)$$
4 8 8
/ 2/a\\ / 2/a\\ / 2/a\\ 4/a\
|1 - tan |-|| *|1 - tan |-|| 16*|1 - tan |-|| *tan |-|
\ \2// \ \4// \ \4// \2/
----------------------------- + -------------------------
8 8
/ 2/a\\ / 2/a\\
|1 + tan |-|| |1 + tan |-||
\ \4// \ \4//
$$\frac{\left(- \tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{8} \left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4}}{\left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{8}} + \frac{16 \left(- \tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{8} \tan^{4}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{8}}$$
4
/ 4/a\\
| 4*sin |-||
| \2/|
|1 - ---------| 4 8/a\
| 2 | 256*sin (a)*sin |-|
\ sin (a) / \2/
---------------- + ----------------------
4 4
/ 4/a\\ / 2 4/a\\
| 4*sin |-|| |sin (a) + 4*sin |-||
| \2/| \ \2//
|1 + ---------|
| 2 |
\ sin (a) /
$$\frac{256 \sin^{8}{\left(\frac{a}{2} \right)} \sin^{4}{\left(a \right)}}{\left(4 \sin^{4}{\left(\frac{a}{2} \right)} + \sin^{2}{\left(a \right)}\right)^{4}} + \frac{\left(- \frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right)^{4}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right)^{4}}$$
// 1 for a mod 2*pi = 0\
// 0 for a mod pi = 0\ || |
|| | || 4 |
|< 4/a\ 8/a\ | + |
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\16 \sin^{8}{\left(\frac{a}{2} \right)} \cot^{4}{\left(\frac{a}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{4} \sin^{8}{\left(\frac{a}{2} \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\
// 0 for a mod pi = 0\ || |
|| | || 4 |
|< 4/a\ 4/a\ | + |
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\16 \sin^{4}{\left(\frac{a}{2} \right)} \cos^{4}{\left(\frac{a}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4} \cos^{8}{\left(\frac{a}{2} \right)} & \text{otherwise} \end{cases}\right)$$
4
/ 4/a\\
| 4*sin |-||
| \2/|
|1 - ---------| 8/a\
| 2 | 256*sin |-|
\ sin (a) / \2/
---------------- + ------------------------
4 4
/ 4/a\\ / 4/a\\
| 4*sin |-|| | 4*sin |-||
| \2/| | \2/| 4
|1 + ---------| |1 + ---------| *sin (a)
| 2 | | 2 |
\ sin (a) / \ sin (a) /
$$\frac{\left(- \frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right)^{4}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right)^{4}} + \frac{256 \sin^{8}{\left(\frac{a}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right)^{4} \sin^{4}{\left(a \right)}}$$
// 0 for a mod pi = 0\
|| | // 1 for a mod 2*pi = 0\
|| 8/a\ | || |
||16*sin |-| | || 4 |
|< \2/ | + |
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{16 \sin^{8}{\left(\frac{a}{2} \right)}}{\tan^{4}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4} \cos^{8}{\left(\frac{a}{2} \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\
|| |
// 0 for a mod pi = 0\ || 4 8/a\ |
|| | ||16*cos (a)*sin |-| |
|< 4/a\ 4/a\ | + |< \2/ |
||16*cos |-|*sin |-| otherwise | ||------------------ otherwise |
\\ \2/ \2/ / || 4 |
|| (-1 + cos(a)) |
\\ /
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\16 \sin^{4}{\left(\frac{a}{2} \right)} \cos^{4}{\left(\frac{a}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{16 \sin^{8}{\left(\frac{a}{2} \right)} \cos^{4}{\left(a \right)}}{\left(\cos{\left(a \right)} - 1\right)^{4}} & \text{otherwise} \end{cases}\right)$$
// / pi\ \
// 0 for a mod pi = 0\ || 0 for |a + --| mod pi = 0|
|| | || \ 2 / |
|< 4/a\ 8/a\ | + |< |
||16*cot |-|*sin |-| otherwise | || 4 4/a pi\ |
\\ \2/ \2/ / ||(1 + sin(a)) *cot |- + --| otherwise |
\\ \2 4 / /
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\16 \sin^{8}{\left(\frac{a}{2} \right)} \cot^{4}{\left(\frac{a}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(a \right)} + 1\right)^{4} \cot^{4}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\ || | || | || 4/a\ | || 4 | || 16*cot |-| | ||/ 2/a\\ | || \2/ | |||-1 + cot |-|| | |<-------------- otherwise | + |<\ \2// | || 4 | ||--------------- otherwise | ||/ 2/a\\ | || 4 | |||1 + cot |-|| | || / 2/a\\ | ||\ \2// | || |1 + cot |-|| | \\ / \\ \ \2// /
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{16 \cot^{4}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{4}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4}} & \text{otherwise} \end{cases}\right)$$
// / 3*pi\ \
|| 1 for |a + ----| mod 2*pi = 0|
|| \ 2 / |
// 1 for a mod 2*pi = 0\ || |
|| | || 4/a\ |
|| 4 | || 16*tan |-| |
|
$$\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{4} \sin^{8}{\left(\frac{a}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{16 \tan^{4}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4}} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\
|| |
// 0 for a mod pi = 0\ || 4 |
|| | ||/ 1 \ |
|| 16 | |||-1 + -------| |
||---------------------- otherwise | ||| 2/a\| |
|| 4 | ||| tan |-|| |
|
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{16}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right)^{4} \tan^{4}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right)^{4}}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right)^{4}} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\
|| |
|| 4 |
// 0 for a mod pi = 0\ ||/ 2/a\ \ |
|| | ||| cos |-| | |
|< 4/a\ 4/a pi\ | + |<| \2/ | 8/a pi\ |
||16*cos |-|*cos |- - --| otherwise | |||-1 + ------------| *cos |- - --| otherwise |
\\ \2/ \2 2 / / ||| 2/a pi\| \2 2 / |
||| cos |- - --|| |
||\ \2 2 // |
\\ /
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\16 \cos^{4}{\left(\frac{a}{2} \right)} \cos^{4}{\left(\frac{a}{2} - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - 1\right)^{4} \cos^{8}{\left(\frac{a}{2} - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\
|| |
|| 4 |
||/ 2/a\ \ |
// 0 for a mod pi = 0\ ||| csc |-| | |
|| | ||| \2/ | |
|| 16 | |||-1 + ------------| |
|<-------------------- otherwise | + |<| 2/pi a\| |
|| 4/a\ 4/pi a\ | ||| csc |-- - -|| |
||csc |-|*csc |-- - -| | ||\ \2 2// |
\\ \2/ \2 2/ / ||-------------------- otherwise |
|| 8/a\ |
|| csc |-| |
|| \2/ |
\\ /
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{16}{\csc^{4}{\left(\frac{a}{2} \right)} \csc^{4}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} - 1\right)^{4}}{\csc^{8}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// / pi\ \
// 0 for a mod pi = 0\ || 0 for |a + --| mod pi = 0|
|| | || \ 2 / |
|| 4/a\ | || |
|| 16*cot |-| | || 4/a pi\ |
|| \2/ | || 16*cot |- + --| |
|<-------------- otherwise | + |< \2 4 / |
|| 4 | ||------------------- otherwise |
||/ 2/a\\ | || 4 |
|||1 + cot |-|| | ||/ 2/a pi\\ |
||\ \2// | |||1 + cot |- + --|| |
\\ / ||\ \2 4 // |
\\ /
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{16 \cot^{4}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{16 \cot^{4}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{4}} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\ || | || | || 16/a\ 8/a\ | || 4 | ||4096*cos |-|*tan |-| | || / 1 \ 16/a\ 8/a\ | |< \4/ \4/ | + |<256*|-1 + -------| *cos |-|*tan |-| otherwise | ||--------------------- otherwise | || | 2/a\| \4/ \4/ | || 4/a\ | || | tan |-|| | || tan |-| | || \ \2// | \\ \2/ / \\ /
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{4096 \cos^{16}{\left(\frac{a}{4} \right)} \tan^{8}{\left(\frac{a}{4} \right)}}{\tan^{4}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\256 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right)^{4} \cos^{16}{\left(\frac{a}{4} \right)} \tan^{8}{\left(\frac{a}{4} \right)} & \text{otherwise} \end{cases}\right)$$
4
/ 2/a pi\\
| cos |- - --||
| \2 2 /|
|1 - ------------|
| 2/a\ | 4/a pi\
| cos |-| | 16*cos |- - --|
\ \2/ / \2 2 /
------------------- + ---------------------------
4 4
/ 2/a pi\\ / 2/a pi\\
| cos |- - --|| | cos |- - --||
| \2 2 /| | \2 2 /| 4/a\
|1 + ------------| |1 + ------------| *cos |-|
| 2/a\ | | 2/a\ | \2/
| cos |-| | | cos |-| |
\ \2/ / \ \2/ /
$$\frac{\left(1 - \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right)^{4}}{\left(1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right)^{4}} + \frac{16 \cos^{4}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right)^{4} \cos^{4}{\left(\frac{a}{2} \right)}}$$
4
/ 2/a\ \
| sec |-| |
| \2/ |
|1 - ------------|
| 2/a pi\| 4/a\
| sec |- - --|| 16*sec |-|
\ \2 2 // \2/
------------------- + --------------------------------
4 4
/ 2/a\ \ / 2/a\ \
| sec |-| | | sec |-| |
| \2/ | | \2/ | 4/a pi\
|1 + ------------| |1 + ------------| *sec |- - --|
| 2/a pi\| | 2/a pi\| \2 2 /
| sec |- - --|| | sec |- - --||
\ \2 2 // \ \2 2 //
$$\frac{\left(- \frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)^{4}}{\left(\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)^{4}} + \frac{16 \sec^{4}{\left(\frac{a}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)^{4} \sec^{4}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}$$
// 1 for a mod 2*pi = 0\
|| |
|| 4 |
||/ 2/a pi\\ |
// 0 for a mod pi = 0\ ||| sec |- - --|| |
|| | ||| \2 2 /| |
|| 16 | |||-1 + ------------| |
|<-------------------- otherwise | + |<| 2/a\ | |
|| 4/a\ 4/a pi\ | ||| sec |-| | |
||sec |-|*sec |- - --| | ||\ \2/ / |
\\ \2/ \2 2 / / ||-------------------- otherwise |
|| 8/a pi\ |
|| sec |- - --| |
|| \2 2 / |
\\ /
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{16}{\sec^{4}{\left(\frac{a}{2} \right)} \sec^{4}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right)^{4}}{\sec^{8}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
4
/ 2/pi a\\
| csc |-- - -||
| \2 2/|
|1 - ------------|
| 2/a\ | 4/pi a\
| csc |-| | 16*csc |-- - -|
\ \2/ / \2 2/
------------------- + ---------------------------
4 4
/ 2/pi a\\ / 2/pi a\\
| csc |-- - -|| | csc |-- - -||
| \2 2/| | \2 2/| 4/a\
|1 + ------------| |1 + ------------| *csc |-|
| 2/a\ | | 2/a\ | \2/
| csc |-| | | csc |-| |
\ \2/ / \ \2/ /
$$\frac{\left(1 - \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right)^{4}}{\left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right)^{4}} + \frac{16 \csc^{4}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right)^{4} \csc^{4}{\left(\frac{a}{2} \right)}}$$
// / 3*pi\ \
// 1 for a mod 2*pi = 0\ || 1 for |a + ----| mod 2*pi = 0|
|| | || \ 2 / |
|| 4 | || |
||/ 2/a\\ | || 4 |
|||-1 + cot |-|| | ||/ 2/a pi\\ |
|<\ \2// | + |<|-1 + tan |- + --|| |
||--------------- otherwise | ||\ \2 4 // |
|| 4 | ||-------------------- otherwise |
|| / 2/a\\ | || 4 |
|| |1 + cot |-|| | ||/ 2/a pi\\ |
\\ \ \2// / |||1 + tan |- + --|| |
\\\ \2 4 // /
$$\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{4}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1\right)^{4}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{4}} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\
// 0 for a mod pi = 0\ || |
|| | || 4 |
|| 8/a\ | || / 1 \ 8/a\ |
|| 4096*tan |-| | ||256*|-1 + -------| *tan |-| |
|| \4/ | || | 2/a\| \4/ |
|<---------------------- otherwise | + |< | tan |-|| |
|| 8 | || \ \2// |
||/ 2/a\\ 4/a\ | ||--------------------------- otherwise |
|||1 + tan |-|| *tan |-| | || 8 |
||\ \4// \2/ | || / 2/a\\ |
\\ / || |1 + tan |-|| |
\\ \ \4// /
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{4096 \tan^{8}{\left(\frac{a}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{8} \tan^{4}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{256 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right)^{4} \tan^{8}{\left(\frac{a}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{8}} & \text{otherwise} \end{cases}\right)$$
// a \ // a \
|| 1 for - mod 2*pi = 0| || 1 for - mod 2*pi = 0|
4 || 2 | || 2 |
/ 2/a\\ || | 4/a\ || |
|1 - tan |-|| *|< 8 | + 16*tan |-|*|< 8 |
\ \2// ||/ 2/a\\ 16/a\ | \2/ ||/ 2/a\\ 16/a\ |
|||-1 + cot |-|| *sin |-| otherwise | |||-1 + cot |-|| *sin |-| otherwise |
||\ \4// \4/ | ||\ \4// \4/ |
\\ / \\ /
$$\left(\left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4} \left(\begin{cases} 1 & \text{for}\: \frac{a}{2} \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{8} \sin^{16}{\left(\frac{a}{4} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(16 \left(\begin{cases} 1 & \text{for}\: \frac{a}{2} \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{8} \sin^{16}{\left(\frac{a}{4} \right)} & \text{otherwise} \end{cases}\right) \tan^{4}{\left(\frac{a}{2} \right)}\right)$$
// 0 for a mod pi = 0\
|| | // 1 for a mod 2*pi = 0\
|| 4 | || |
|| sin (a) | || 4 |
||------------------------ otherwise | ||/ 2 4/a\\ |
|| 4 | |||sin (a) - 4*sin |-|| |
|
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{\sin^{4}{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right)^{4} \sin^{8}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(- 4 \sin^{4}{\left(\frac{a}{2} \right)} + \sin^{2}{\left(a \right)}\right)^{4}}{\left(4 \sin^{4}{\left(\frac{a}{2} \right)} + \sin^{2}{\left(a \right)}\right)^{4}} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\
|| |
// 0 for a mod pi = 0\ || 4 |
|| | ||/ 2 \ |
|| 4 | ||| sin (a) | |
|| sin (a) | |||-1 + ---------| |
||------------------------ otherwise | ||| 4/a\| |
|| 4 | ||| 4*sin |-|| |
|
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{\sin^{4}{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right)^{4} \sin^{8}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right)^{4}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right)^{4}} & \text{otherwise} \end{cases}\right)$$
// a \
|| 1 for - mod 2*pi = 0|
|| 2 |
|| |
|| 8 |
||/ 2/a\\ |
16*|<|-1 + cot |-|| |
// a \ ||\ \4// |
|| 1 for - mod 2*pi = 0| ||--------------- otherwise |
|| 2 | || 8 |
|| | || / 2/a\\ |
4 || 8 | || |1 + cot |-|| |
/ 1 \ ||/ 2/a\\ | \\ \ \4// /
|1 - -------| *|<|-1 + cot |-|| | + -----------------------------------------
| 2/a\| ||\ \4// | 4/a\
| cot |-|| ||--------------- otherwise | cot |-|
\ \2// || 8 | \2/
|| / 2/a\\ |
|| |1 + cot |-|| |
\\ \ \4// /
$$\left(\left(1 - \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right)^{4} \left(\begin{cases} 1 & \text{for}\: \frac{a}{2} \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{8}}{\left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{8}} & \text{otherwise} \end{cases}\right)\right) + \left(\frac{16 \left(\begin{cases} 1 & \text{for}\: \frac{a}{2} \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{8}}{\left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{8}} & \text{otherwise} \end{cases}\right)}{\cot^{4}{\left(\frac{a}{2} \right)}}\right)$$
// 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\
|| | || |
|| // a \ | || // a \ |
|| || 0 for - mod pi = 0| | || 4 || 0 for - mod pi = 0| |
|< 4/a\ || 2 | | + |
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\16 \left(\begin{cases} 0 & \text{for}\: \frac{a}{2} \bmod \pi = 0 \\256 \sin^{16}{\left(\frac{a}{4} \right)} \cot^{8}{\left(\frac{a}{4} \right)} & \text{otherwise} \end{cases}\right) \cot^{4}{\left(\frac{a}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{4} \left(\begin{cases} 0 & \text{for}\: \frac{a}{2} \bmod \pi = 0 \\256 \sin^{16}{\left(\frac{a}{4} \right)} \cot^{8}{\left(\frac{a}{4} \right)} & \text{otherwise} \end{cases}\right) & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\
|| |
// 0 for a mod pi = 0\ || 4 |
|| | ||/ 2/a\ \ |
|| 4/a\ | ||| cos |-| | |
|| 16*cos |-| | ||| \2/ | |
|| \2/ | |||-1 + ------------| |
||-------------------------------- otherwise | ||| 2/a pi\| |
|| 4 | ||| cos |- - --|| |
|
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{16 \cos^{4}{\left(\frac{a}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)^{4} \cos^{4}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - 1\right)^{4}}{\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)^{4}} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\
|| |
// 0 for a mod pi = 0\ || 4 |
|| | ||/ 2/a pi\\ |
|| 4/a pi\ | ||| sec |- - --|| |
|| 16*sec |- - --| | ||| \2 2 /| |
|| \2 2 / | |||-1 + ------------| |
||--------------------------- otherwise | ||| 2/a\ | |
|| 4 | ||| sec |-| | |
|
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{16 \sec^{4}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right)^{4} \sec^{4}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right)^{4}}{\left(1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right)^{4}} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\
|| |
// 0 for a mod pi = 0\ || 4 |
|| | ||/ 2/a\ \ |
|| 4/a\ | ||| csc |-| | |
|| 16*csc |-| | ||| \2/ | |
|| \2/ | |||-1 + ------------| |
||-------------------------------- otherwise | ||| 2/pi a\| |
|| 4 | ||| csc |-- - -|| |
|
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{16 \csc^{4}{\left(\frac{a}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1\right)^{4} \csc^{4}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} - 1\right)^{4}}{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1\right)^{4}} & \text{otherwise} \end{cases}\right)$$
// 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| |
|| || 2 | | || || 2 | |
|| || | | || || | |
|| || 8/a\ | | || 4 || 8/a\ | |
|< 4/a\ || 256*cot |-| | | + |
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\16 \left(\begin{cases} 0 & \text{for}\: \frac{a}{2} \bmod \pi = 0 \\\frac{256 \cot^{8}{\left(\frac{a}{4} \right)}}{\left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{8}} & \text{otherwise} \end{cases}\right) \cot^{4}{\left(\frac{a}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{4} \left(\begin{cases} 0 & \text{for}\: \frac{a}{2} \bmod \pi = 0 \\\frac{256 \cot^{8}{\left(\frac{a}{4} \right)}}{\left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{8}} & \text{otherwise} \end{cases}\right) & \text{otherwise} \end{cases}\right)$$
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)), ((-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))