Господин Экзамен
Другие калькуляторы
-
Как пользоваться?
- sin(x)^4+2*sin(x)^2*cos(x)^2+cos(x)^4 если x=1/2
- 3*sqrt(4*sqrt(4*m^6)) если m=1/3
- (a-7)*(a+7)/(a-7)*(a+7) если a=-1
- (-c^3)^2*12*c^6 если c=1/2
- Разложить многочлен на множители:
- 3*x^2-12
- 16-49*y^2
- x^2-12*x+36
- 20*n^2-35*a-14*a*n+50*n
- Общий знаменатель:
- a/4*c-a^2+16*c^2/4*a*c+4*c-a/a
- 2*m*n/(m^3+n^3)-2*m/(m^2-n^2)-1/(m-n)
- (m+7/m-n+7/n)*m*n/m^2-n^2
- 4/b-5+1/b
- Умножение столбиком:
- 70*12
- 17*17
- 15*30
- 50*10
- Деление столбиком:
- 6083/4
- 612/7
- 150/2
- 70/8
- Раскрыть скобки в:
- a*(a+1)-(a-3)^2
- (m-1)*(m+1)-(m-3)^2
- (4*m^2+6)*(4*m-6)
- (4*a^2+3)*(2-b)
- Разложение числа на простые множители:
- 4982
- 4356
- 2340
- 2520
- График функции y =:
- 1/2
- Производная:
- 1/2
- Интеграл d{x}:
-
1/2
-
Идентичные выражения
- sin(x)^ четыре + два *sin(x)^ два *cos(x)^ два +cos(x)^ четыре если x= один / два
- синус от (x) в степени 4 плюс 2 умножить на синус от (x) в квадрате умножить на косинус от (x) в квадрате плюс косинус от (x) в степени 4 если x равно 1 делить на 2
- синус от (x) в степени четыре плюс два умножить на синус от (x) в степени два умножить на косинус от (x) в степени два плюс косинус от (x) в степени четыре если x равно один делить на два
- sin(x)4+2*sin(x)2*cos(x)2+cos(x)4 если x=1/2
- sinx4+2*sinx2*cosx2+cosx4 если x=1/2
- sin(x)⁴+2*sin(x)²*cos(x)²+cos(x)⁴ если x=1/2
- sin(x) в степени 4+2*sin(x) в степени 2*cos(x) в степени 2+cos(x) в степени 4 если x=1/2
- sin(x)^4+2sin(x)^2cos(x)^2+cos(x)^4 если x=1/2
- sin(x)4+2sin(x)2cos(x)2+cos(x)4 если x=1/2
- sinx4+2sinx2cosx2+cosx4 если x=1/2
- sinx^4+2sinx^2cosx^2+cosx^4 если x=1/2
- sin(x)^4+2*sin(x)^2*cos(x)^2+cos(x)^4 при x=1/2
- sin(x)^4+2*sin(x)^2*cos(x)^2+cos(x)^4 если x=1 разделить на 2
-
Похожие выражения
- sin(x)^4+2*sin(x)^2*cos(x)^2-cos(x)^4 если x=1/2
- sin(x)^4-2*sin(x)^2*cos(x)^2+cos(x)^4 если x=1/2
- sinx^4+2*sinx^2*cosx^2+cosx^4 если x=1/2
sin(x)^4+2*sin(x)^2*cos(x)^2+cos(x)^4 если x=1/2
Решение
Вы ввели
[src]
4 2 2 4 sin (x) + 2*sin (x)*cos (x) + cos (x)
$$\sin^{4}{\left(x \right)} + 2 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} + \cos^{4}{\left(x \right)}$$
sin(x)^4 + 2*sin(x)^2*cos(x)^2 + cos(x)^4
Степени
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2
/ I*x -I*x\ 2
4 4 |e e | / -I*x I*x\
/ I*x -I*x\ / -I*x I*x\ |---- + -----| *\- e + e /
|e e | \- e + e / \ 2 2 /
|---- + -----| + ----------------- - ---------------------------------
\ 2 2 / 16 2
$$\left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{4} - \frac{\left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{2} \left(e^{i x} - e^{- i x}\right)^{2}}{2} + \frac{\left(e^{i x} - e^{- i x}\right)^{4}}{16}$$
(exp(i*x)/2 + exp(-i*x)/2)^4 + (-exp(-i*x) + exp(i*x))^4/16 - (exp(i*x)/2 + exp(-i*x)/2)^2*(-exp(-i*x) + exp(i*x))^2/2
Численный ответ
[src]
cos(x)^4 + sin(x)^4 + 2.0*cos(x)^2*sin(x)^2
cos(x)^4 + sin(x)^4 + 2.0*cos(x)^2*sin(x)^2
Комбинаторика
[src]
2 / 2 2 \ \cos (x) + sin (x)/
$$\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)^{2}$$
(cos(x)^2 + sin(x)^2)^2
Тригонометрическая часть
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1
$$1$$
1 1 2 ------- + ------- + --------------- 4 4 2 2 csc (x) sec (x) csc (x)*sec (x)
$$\frac{1}{\sec^{4}{\left(x \right)}} + \frac{2}{\csc^{2}{\left(x \right)} \sec^{2}{\left(x \right)}} + \frac{1}{\csc^{4}{\left(x \right)}}$$
4 4/ pi\ 2 2/ pi\
cos (x) + cos |x - --| + 2*cos (x)*cos |x - --|
\ 2 / \ 2 /
$$\cos^{4}{\left(x \right)} + 2 \cos^{2}{\left(x \right)} \cos^{2}{\left(x - \frac{\pi}{2} \right)} + \cos^{4}{\left(x - \frac{\pi}{2} \right)}$$
4 4/ pi\ 2 2/ pi\
sin (x) + sin |x + --| + 2*sin (x)*sin |x + --|
\ 2 / \ 2 /
$$\sin^{4}{\left(x \right)} + 2 \sin^{2}{\left(x \right)} \sin^{2}{\left(x + \frac{\pi}{2} \right)} + \sin^{4}{\left(x + \frac{\pi}{2} \right)}$$
1 1 2
------- + ------------ + --------------------
4 4/ pi\ 2 2/ pi\
sec (x) sec |x - --| sec (x)*sec |x - --|
\ 2 / \ 2 /
$$\frac{1}{\sec^{4}{\left(x - \frac{\pi}{2} \right)}} + \frac{2}{\sec^{2}{\left(x \right)} \sec^{2}{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{\sec^{4}{\left(x \right)}}$$
1 1 2
------- + ------------ + --------------------
4 4/pi \ 2 2/pi \
sec (x) sec |-- - x| sec (x)*sec |-- - x|
\2 / \2 /
$$\frac{1}{\sec^{4}{\left(- x + \frac{\pi}{2} \right)}} + \frac{2}{\sec^{2}{\left(x \right)} \sec^{2}{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{\sec^{4}{\left(x \right)}}$$
1 1 2
------- + ------------ + --------------------
4 4/pi \ 2 2/pi \
csc (x) csc |-- - x| csc (x)*csc |-- - x|
\2 / \2 /
$$\frac{1}{\csc^{4}{\left(- x + \frac{\pi}{2} \right)}} + \frac{2}{\csc^{2}{\left(x \right)} \csc^{2}{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{\csc^{4}{\left(x \right)}}$$
3 cos(4*x) /1 cos(2*x)\ /1 cos(2*x)\ - + -------- + 2*|- + --------|*|- - --------| 4 4 \2 2 / \2 2 /
$$2 \cdot \left(- \frac{\cos{\left(2 x \right)}}{2} + \frac{1}{2}\right) \left(\frac{\cos{\left(2 x \right)}}{2} + \frac{1}{2}\right) + \frac{\cos{\left(4 x \right)}}{4} + \frac{3}{4}$$
1 1 2
------------ + ------------ + -------------------------
4 4/pi \ 2 2/pi \
csc (pi - x) csc |-- - x| csc (pi - x)*csc |-- - x|
\2 / \2 /
$$\frac{1}{\csc^{4}{\left(- x + \frac{\pi}{2} \right)}} + \frac{2}{\csc^{2}{\left(- x + \pi \right)} \csc^{2}{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{\csc^{4}{\left(- x + \pi \right)}}$$
4 4 4/x\ 2 2/x\
cos (x) + (1 + cos(x)) *tan |-| + 4*cos (x)*sin |-|*(1 + cos(x))
\2/ \2/
$$\left(\cos{\left(x \right)} + 1\right)^{4} \tan^{4}{\left(\frac{x}{2} \right)} + 4 \left(\cos{\left(x \right)} + 1\right) \sin^{2}{\left(\frac{x}{2} \right)} \cos^{2}{\left(x \right)} + \cos^{4}{\left(x \right)}$$
2
/ 2 2 \ / 2 2 \ / 2 2 \
3 \cos (x) - sin (x)/ 2 2 |1 cos (x) sin (x)| |1 sin (x) cos (x)|
- + -------------------- - cos (x)*sin (x) + 2*|- + ------- - -------|*|- + ------- - -------|
4 4 \2 2 2 / \2 2 2 /
$$- \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} + \frac{\left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)^{2}}{4} + 2 \left(- \frac{\sin^{2}{\left(x \right)}}{2} + \frac{\cos^{2}{\left(x \right)}}{2} + \frac{1}{2}\right) \left(\frac{\sin^{2}{\left(x \right)}}{2} - \frac{\cos^{2}{\left(x \right)}}{2} + \frac{1}{2}\right) + \frac{3}{4}$$
4 2
/ 2/x\\ 4/x\ / 2/x\\ 2/x\
|1 - tan |-|| 16*tan |-| 8*|1 - tan |-|| *tan |-|
\ \2// \2/ \ \2// \2/
-------------- + -------------- + ------------------------
4 4 4
/ 2/x\\ / 2/x\\ / 2/x\\
|1 + tan |-|| |1 + tan |-|| |1 + tan |-||
\ \2// \ \2// \ \2//
$$\frac{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}} + \frac{8 \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}} + \frac{16 \tan^{4}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}}$$
4/x pi\
16*tan |- + --| 2 2
\2 4 / 4/x\ 8/x\ / /x\\ / /x\\ 4/x\
------------------- + 16*cot |-|*sin |-| + |1 + tan|-|| *|-1 + tan|-|| *cos |-|*(1 - cos(2*x))
4 \2/ \2/ \ \2// \ \2// \2/
/ 2/x pi\\
|1 + tan |- + --||
\ \2 4 //
$$16 \sin^{8}{\left(\frac{x}{2} \right)} \cot^{4}{\left(\frac{x}{2} \right)} + \left(- \cos{\left(2 x \right)} + 1\right) \left(\tan{\left(\frac{x}{2} \right)} - 1\right)^{2} \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2} \cos^{4}{\left(\frac{x}{2} \right)} + \frac{16 \tan^{4}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{4}}$$
4/x pi\
16*tan |- + --| 2 2
\2 4 / 8/x\ 4/x\ / /x\\ / /x\\ 4/x\
------------------- + 16*cos |-|*tan |-| + |1 + tan|-|| *|-1 + tan|-|| *cos |-|*(1 - cos(2*x))
4 \2/ \2/ \ \2// \ \2// \2/
/ 2/x pi\\
|1 + tan |- + --||
\ \2 4 //
$$16 \cos^{8}{\left(\frac{x}{2} \right)} \tan^{4}{\left(\frac{x}{2} \right)} + \left(- \cos{\left(2 x \right)} + 1\right) \left(\tan{\left(\frac{x}{2} \right)} - 1\right)^{2} \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2} \cos^{4}{\left(\frac{x}{2} \right)} + \frac{16 \tan^{4}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{4}}$$
4 2
/ 1 \ / 1 \
|1 - -------| 8*|1 - -------|
| 2/x\| | 2/x\|
| cot |-|| | cot |-||
\ \2// 16 \ \2//
-------------- + ---------------------- + ----------------------
4 4 4
/ 1 \ / 1 \ 4/x\ / 1 \ 2/x\
|1 + -------| |1 + -------| *cot |-| |1 + -------| *cot |-|
| 2/x\| | 2/x\| \2/ | 2/x\| \2/
| cot |-|| | cot |-|| | cot |-||
\ \2// \ \2// \ \2//
$$\frac{\left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{4}}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{4}} + \frac{8 \left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{4} \cot^{2}{\left(\frac{x}{2} \right)}} + \frac{16}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{4} \cot^{4}{\left(\frac{x}{2} \right)}}$$
2 2
4/x\ / 2/x\\ / 2/x pi\\ 4/x\
4 16*tan |-| 2*|-1 + cot |-|| *|-1 + tan |- + --|| *sin |-|
/ 2/x\\ 8/x\ \2/ \ \2// \ \2 4 // \2/
|-1 + cot |-|| *sin |-| + -------------- + ----------------------------------------------
\ \2// \2/ 4 2
/ 2/x\\ / 2/x pi\\
|1 + tan |-|| |1 + tan |- + --||
\ \2// \ \2 4 //
$$\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{4} \sin^{8}{\left(\frac{x}{2} \right)} + \frac{2 \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2} \sin^{4}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} + \frac{16 \tan^{4}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}}$$
4 2 2
/ 2/x pi\\ 4 / 2/x pi\\ / 2/x\\ 2 4/x\
4 |1 - cot |- + --|| *(1 + sin(x)) |1 - cot |- + --|| *|1 - tan |-|| *(1 + sin(x)) *cos |-|
/ 2/x\\ 8/x\ \ \2 4 // \ \2 4 // \ \2// \2/
|1 - tan |-|| *cos |-| + --------------------------------- + --------------------------------------------------------
\ \2// \2/ 16 2
$$\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4} \cos^{8}{\left(\frac{x}{2} \right)} + \frac{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \left(- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\sin{\left(x \right)} + 1\right)^{2} \cos^{4}{\left(\frac{x}{2} \right)}}{2} + \frac{\left(- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{4} \left(\sin{\left(x \right)} + 1\right)^{4}}{16}$$
4/x\ 4/x pi\ 2/x\ 2/x pi\
16*cot |-| 16*tan |- + --| 32*cot |-|*tan |- + --|
\2/ \2 4 / \2/ \2 4 /
-------------- + ------------------- + ----------------------------------
4 4 2 2
/ 2/x\\ / 2/x pi\\ / 2/x\\ / 2/x pi\\
|1 + cot |-|| |1 + tan |- + --|| |1 + cot |-|| *|1 + tan |- + --||
\ \2// \ \2 4 // \ \2// \ \2 4 //
$$\frac{16 \cot^{4}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}} + \frac{32 \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + \frac{16 \tan^{4}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{4}}$$
4/x\ 4/x pi\ 2/x\ 2/x pi\
16*tan |-| 16*tan |- + --| 32*tan |-|*tan |- + --|
\2/ \2 4 / \2/ \2 4 /
-------------- + ------------------- + ----------------------------------
4 4 2 2
/ 2/x\\ / 2/x pi\\ / 2/x\\ / 2/x pi\\
|1 + tan |-|| |1 + tan |- + --|| |1 + tan |-|| *|1 + tan |- + --||
\ \2// \ \2 4 // \ \2// \ \2 4 //
$$\frac{16 \tan^{4}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{4}} + \frac{32 \tan^{2}{\left(\frac{x}{2} \right)} \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} + \frac{16 \tan^{4}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}}$$
4 4 2 2
/ 2/x\\ / 2/x pi\\ / 2/x\\ / 2/x pi\\
|-1 + cot |-|| |-1 + tan |- + --|| 2*|-1 + cot |-|| *|-1 + tan |- + --||
\ \2// \ \2 4 // \ \2// \ \2 4 //
--------------- + -------------------- + --------------------------------------
4 4 2 2
/ 2/x\\ / 2/x pi\\ / 2/x\\ / 2/x pi\\
|1 + cot |-|| |1 + tan |- + --|| |1 + cot |-|| *|1 + tan |- + --||
\ \2// \ \2 4 // \ \2// \ \2 4 //
$$\frac{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)^{4}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{4}} + \frac{2 \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + \frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{4}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}}$$
4 4 2 2
/ 2/x pi\\ / 2/x\\ / 2/x pi\\ / 2/x\\
|1 - cot |- + --|| |1 - tan |-|| 2*|1 - cot |- + --|| *|1 - tan |-||
\ \2 4 // \ \2// \ \2 4 // \ \2//
------------------- + -------------- + ------------------------------------
4 4 2 2
/ 2/x pi\\ / 2/x\\ / 2/x pi\\ / 2/x\\
|1 + cot |- + --|| |1 + tan |-|| |1 + cot |- + --|| *|1 + tan |-||
\ \2 4 // \ \2// \ \2 4 // \ \2//
$$\frac{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}} + \frac{2 \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \left(- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} + \frac{\left(- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{4}}{\left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{4}}$$
// 0 for x mod pi = 0\
|| |
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ || 4 | // 1 for x mod 2*pi = 0\
|| | || | ||(-1 + cos(x)) | || |
2*|< 2 |*|< 2 | + |<-------------- otherwise | + |< 4 |
||sin (x) otherwise | ||cos (x) otherwise | || 4/x\ | ||cos (x) otherwise |
\\ / \\ / || tan |-| | \\ /
|| \2/ |
\\ /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{\left(\cos{\left(x \right)} - 1\right)^{4}}{\tan^{4}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{4}{\left(x \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\
|| |
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ // 0 for x mod pi = 0\ || 4 |
|| | || | || | ||/ 2 \ |
2*|< 2 |*|< 2 | + |< 4 | + |<| sin (x) | 8/x\ |
||sin (x) otherwise | ||cos (x) otherwise | ||sin (x) otherwise | |||-1 + ---------| *sin |-| otherwise |
\\ / \\ / \\ / ||| 4/x\| \2/ |
||| 4*sin |-|| |
\\\ \2// /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{4}{\left(x \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\left(-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{4} \sin^{8}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ // 0 for x mod pi = 0\ || |
|| | || | || | || 4 |
2*|< 2 |*|< 2 | + |< 4/x\ 8/x\ | + |
$$\left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\16 \sin^{8}{\left(\frac{x}{2} \right)} \cot^{4}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{4} \sin^{8}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ // 0 for x mod pi = 0\ || |
|| | || | || | || 4 |
2*|< 2 |*|< 2 | + |< 4/x\ 4/x\ | + |
$$\left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\16 \sin^{4}{\left(\frac{x}{2} \right)} \cos^{4}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4} \cos^{8}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\
|| |
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ // 0 for x mod pi = 0\ || 4 |
|| | || | || | ||/ 2 \ |
2*|< 2 |*|< 2/ pi\ | + |< 4 | + |<| sin (x) | 8/x\ |
||sin (x) otherwise | ||sin |x + --| otherwise | ||sin (x) otherwise | |||-1 + ---------| *sin |-| otherwise |
\\ / \\ \ 2 / / \\ / ||| 4/x\| \2/ |
||| 4*sin |-|| |
\\\ \2// /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\sin^{2}{\left(x + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{4}{\left(x \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\left(-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{4} \sin^{8}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right)$$
// 0 for x mod pi = 0\
|| | // 1 for x mod 2*pi = 0\
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ || 8/x\ | || |
|| | || | ||16*sin |-| | || 4 |
2*|< 2 |*|< 2 | + |< \2/ | + |
$$\left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{16 \sin^{8}{\left(\frac{x}{2} \right)}}{\tan^{4}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4} \cos^{8}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right)$$
4 2
/ 4/x\\ / 4/x\\
| 4*sin |-|| | 4*sin |-||
| \2/| | \2/| 4/x\
|1 - ---------| 8/x\ 32*|1 - ---------| *sin |-|
| 2 | 256*sin |-| | 2 | \2/
\ sin (x) / \2/ \ sin (x) /
---------------- + ------------------------ + ---------------------------
4 4 4
/ 4/x\\ / 4/x\\ / 4/x\\
| 4*sin |-|| | 4*sin |-|| | 4*sin |-||
| \2/| | \2/| 4 | \2/| 2
|1 + ---------| |1 + ---------| *sin (x) |1 + ---------| *sin (x)
| 2 | | 2 | | 2 |
\ sin (x) / \ sin (x) / \ sin (x) /
$$\frac{\left(- \frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{4}}{\left(\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{4}} + \frac{32 \left(- \frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{2} \sin^{4}{\left(\frac{x}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{4} \sin^{2}{\left(x \right)}} + \frac{256 \sin^{8}{\left(\frac{x}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{4} \sin^{4}{\left(x \right)}}$$
// 1 for x mod 2*pi = 0\
|| |
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ // 0 for x mod pi = 0\ || 4 8/x\ |
|| | || | || | ||16*cos (x)*sin |-| |
2*|< 2 |*|< 2 | + |< 4/x\ 4/x\ | + |< \2/ |
||sin (x) otherwise | ||cos (x) otherwise | ||16*cos |-|*sin |-| otherwise | ||------------------ otherwise |
\\ / \\ / \\ \2/ \2/ / || 4 |
|| (-1 + cos(x)) |
\\ /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\16 \sin^{4}{\left(\frac{x}{2} \right)} \cos^{4}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{16 \sin^{8}{\left(\frac{x}{2} \right)} \cos^{4}{\left(x \right)}}{\left(\cos{\left(x \right)} - 1\right)^{4}} & \text{otherwise} \end{cases}\right)$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\
|| | || |
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ || 16/x\ 8/x\ | || 4 |
|| | || | ||4096*cos |-|*tan |-| | || / 1 \ 16/x\ 8/x\ |
2*|< 2 |*|< 2 | + |< \4/ \4/ | + |<256*|-1 + -------| *cos |-|*tan |-| otherwise |
||sin (x) otherwise | ||cos (x) otherwise | ||--------------------- otherwise | || | 2/x\| \4/ \4/ |
\\ / \\ / || 4/x\ | || | tan |-|| |
|| tan |-| | || \ \2// |
\\ \2/ / \\ /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4096 \cos^{16}{\left(\frac{x}{4} \right)} \tan^{8}{\left(\frac{x}{4} \right)}}{\tan^{4}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\256 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{4} \cos^{16}{\left(\frac{x}{4} \right)} \tan^{8}{\left(\frac{x}{4} \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\
|| |
|| 4 |
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ // 0 for x mod pi = 0\ ||/ 2/x\ \ |
|| | || | || | ||| cos |-| | |
2*|< 2/ pi\ |*|< 2 | + |< 4/x\ 4/x pi\ | + |<| \2/ | 8/x pi\ |
||cos |x - --| otherwise | ||cos (x) otherwise | ||16*cos |-|*cos |- - --| otherwise | |||-1 + ------------| *cos |- - --| otherwise |
\\ \ 2 / / \\ / \\ \2/ \2 2 / / ||| 2/x pi\| \2 2 / |
||| cos |- - --|| |
||\ \2 2 // |
\\ /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\cos^{2}{\left(x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\16 \cos^{4}{\left(\frac{x}{2} \right)} \cos^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1\right)^{4} \cos^{8}{\left(\frac{x}{2} - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)$$
// / pi\ \ // / pi\ \
// 0 for x mod pi = 0\ || 0 for |x + --| mod pi = 0| // 0 for x mod pi = 0\ || 0 for |x + --| mod pi = 0|
|| | || \ 2 / | || | || \ 2 / |
2*|< 2 |*|< | + |< 4/x\ 8/x\ | + |< |
||sin (x) otherwise | || 2 2/x pi\ | ||16*cot |-|*sin |-| otherwise | || 4 4/x pi\ |
\\ / ||(1 + sin(x)) *cot |- + --| otherwise | \\ \2/ \2/ / ||(1 + sin(x)) *cot |- + --| otherwise |
\\ \2 4 / / \\ \2 4 / /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(x \right)} + 1\right)^{2} \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\16 \sin^{8}{\left(\frac{x}{2} \right)} \cot^{4}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(x \right)} + 1\right)^{4} \cot^{4}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\
|| |
|| 4 |
||/ 2/x\ \ |
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ // 0 for x mod pi = 0\ ||| csc |-| | |
|| | || | || | ||| \2/ | |
|| 1 | || 1 | || 16 | |||-1 + ------------| |
2*|<------- otherwise |*|<------------ otherwise | + |<-------------------- otherwise | + |<| 2/pi x\| |
|| 2 | || 2/pi \ | || 4/x\ 4/pi x\ | ||| csc |-- - -|| |
||csc (x) | ||csc |-- - x| | ||csc |-|*csc |-- - -| | ||\ \2 2// |
\\ / \\ \2 / / \\ \2/ \2 2/ / ||-------------------- otherwise |
|| 8/x\ |
|| csc |-| |
|| \2/ |
\\ /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\csc^{2}{\left(x \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{1}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{16}{\csc^{4}{\left(\frac{x}{2} \right)} \csc^{4}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - 1\right)^{4}}{\csc^{8}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\
|| |
|| 4 |
||/ 2/x pi\\ |
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ // 0 for x mod pi = 0\ ||| sec |- - --|| |
|| | || | || | ||| \2 2 /| |
|| 1 | || 1 | || 16 | |||-1 + ------------| |
2*|<------------ otherwise |*|<------- otherwise | + |<-------------------- otherwise | + |<| 2/x\ | |
|| 2/ pi\ | || 2 | || 4/x\ 4/x pi\ | ||| sec |-| | |
||sec |x - --| | ||sec (x) | ||sec |-|*sec |- - --| | ||\ \2/ / |
\\ \ 2 / / \\ / \\ \2/ \2 2 / / ||-------------------- otherwise |
|| 8/x pi\ |
|| sec |- - --| |
|| \2 2 / |
\\ /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{1}{\sec^{2}{\left(x \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{16}{\sec^{4}{\left(\frac{x}{2} \right)} \sec^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{4}}{\sec^{8}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// / 3*pi\ \
|| 1 for |x + ----| mod 2*pi = 0|
|| \ 2 / |
// / 3*pi\ \ // 1 for x mod 2*pi = 0\ || |
// 1 for x mod 2*pi = 0\ || 1 for |x + ----| mod 2*pi = 0| || | || 4/x\ |
|| | || \ 2 / | || 4 | || 16*tan |-| |
2*|< 2 |*|< | + |
$$\left(2 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\- 4 \cos^{4}{\left(\frac{x}{2} \right)} + 4 \cos^{2}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{4} \sin^{8}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{16 \tan^{4}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}} & \text{otherwise} \end{cases}\right)$$
4 2
/ 2/x pi\\ / 2/x pi\\
| cos |- - --|| | cos |- - --||
| \2 2 /| | \2 2 /| 2/x pi\
|1 - ------------| 8*|1 - ------------| *cos |- - --|
| 2/x\ | 4/x pi\ | 2/x\ | \2 2 /
| cos |-| | 16*cos |- - --| | cos |-| |
\ \2/ / \2 2 / \ \2/ /
------------------- + --------------------------- + ----------------------------------
4 4 4
/ 2/x pi\\ / 2/x pi\\ / 2/x pi\\
| cos |- - --|| | cos |- - --|| | cos |- - --||
| \2 2 /| | \2 2 /| 4/x\ | \2 2 /| 2/x\
|1 + ------------| |1 + ------------| *cos |-| |1 + ------------| *cos |-|
| 2/x\ | | 2/x\ | \2/ | 2/x\ | \2/
| cos |-| | | cos |-| | | cos |-| |
\ \2/ / \ \2/ / \ \2/ /
$$\frac{\left(1 - \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{4}}{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{4}} + \frac{8 \left(1 - \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{4} \cos^{2}{\left(\frac{x}{2} \right)}} + \frac{16 \cos^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{4} \cos^{4}{\left(\frac{x}{2} \right)}}$$
4 2
/ 2/x\ \ / 2/x\ \
| sec |-| | | sec |-| |
| \2/ | | \2/ | 2/x\
|1 - ------------| 8*|1 - ------------| *sec |-|
| 2/x pi\| 4/x\ | 2/x pi\| \2/
| sec |- - --|| 16*sec |-| | sec |- - --||
\ \2 2 // \2/ \ \2 2 //
------------------- + -------------------------------- + --------------------------------
4 4 4
/ 2/x\ \ / 2/x\ \ / 2/x\ \
| sec |-| | | sec |-| | | sec |-| |
| \2/ | | \2/ | 4/x pi\ | \2/ | 2/x pi\
|1 + ------------| |1 + ------------| *sec |- - --| |1 + ------------| *sec |- - --|
| 2/x pi\| | 2/x pi\| \2 2 / | 2/x pi\| \2 2 /
| sec |- - --|| | sec |- - --|| | sec |- - --||
\ \2 2 // \ \2 2 // \ \2 2 //
$$\frac{\left(- \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{4}}{\left(\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{4}} + \frac{8 \left(- \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec^{2}{\left(\frac{x}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{4} \sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + \frac{16 \sec^{4}{\left(\frac{x}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{4} \sec^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}$$
4 2
/ 2/pi x\\ / 2/pi x\\
| csc |-- - -|| | csc |-- - -||
| \2 2/| | \2 2/| 2/pi x\
|1 - ------------| 8*|1 - ------------| *csc |-- - -|
| 2/x\ | 4/pi x\ | 2/x\ | \2 2/
| csc |-| | 16*csc |-- - -| | csc |-| |
\ \2/ / \2 2/ \ \2/ /
------------------- + --------------------------- + ----------------------------------
4 4 4
/ 2/pi x\\ / 2/pi x\\ / 2/pi x\\
| csc |-- - -|| | csc |-- - -|| | csc |-- - -||
| \2 2/| | \2 2/| 4/x\ | \2 2/| 2/x\
|1 + ------------| |1 + ------------| *csc |-| |1 + ------------| *csc |-|
| 2/x\ | | 2/x\ | \2/ | 2/x\ | \2/
| csc |-| | | csc |-| | | csc |-| |
\ \2/ / \ \2/ / \ \2/ /
$$\frac{\left(1 - \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{4}}{\left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{4}} + \frac{8 \left(1 - \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{4} \csc^{2}{\left(\frac{x}{2} \right)}} + \frac{16 \csc^{4}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{4} \csc^{4}{\left(\frac{x}{2} \right)}}$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ // 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ || | || | || | || | || 2/x\ | || 2 | || 4/x\ | || 4 | || 4*cot |-| | ||/ 2/x\\ | || 16*cot |-| | ||/ 2/x\\ | || \2/ | |||-1 + cot |-|| | || \2/ | |||-1 + cot |-|| | 2*|<-------------- otherwise |*|<\ \2// | + |<-------------- otherwise | + |<\ \2// | || 2 | ||--------------- otherwise | || 4 | ||--------------- otherwise | ||/ 2/x\\ | || 2 | ||/ 2/x\\ | || 4 | |||1 + cot |-|| | || / 2/x\\ | |||1 + cot |-|| | || / 2/x\\ | ||\ \2// | || |1 + cot |-|| | ||\ \2// | || |1 + cot |-|| | \\ / \\ \ \2// / \\ / \\ \ \2// /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{16 \cot^{4}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{4}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\ // 1 for x mod 2*pi = 0\
|| | || |
// 0 for x mod pi = 0\ || 2 | // 0 for x mod pi = 0\ || 4 |
|| | ||/ 1 \ | || | ||/ 1 \ |
|| 4 | |||-1 + -------| | || 16 | |||-1 + -------| |
||---------------------- otherwise | ||| 2/x\| | ||---------------------- otherwise | ||| 2/x\| |
|| 2 | ||| tan |-|| | || 4 | ||| tan |-|| |
2*|
$$\left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \tan^{2}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{16}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{4} \tan^{4}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{4}}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{4}} & \text{otherwise} \end{cases}\right)$$
// / pi\ \ // / pi\ \
// 0 for x mod pi = 0\ || 0 for |x + --| mod pi = 0| // 0 for x mod pi = 0\ || 0 for |x + --| mod pi = 0|
|| | || \ 2 / | || | || \ 2 / |
|| 2/x\ | || | || 4/x\ | || |
|| 4*cot |-| | || 2/x pi\ | || 16*cot |-| | || 4/x pi\ |
|| \2/ | || 4*cot |- + --| | || \2/ | || 16*cot |- + --| |
2*|<-------------- otherwise |*|< \2 4 / | + |<-------------- otherwise | + |< \2 4 / |
|| 2 | ||------------------- otherwise | || 4 | ||------------------- otherwise |
||/ 2/x\\ | || 2 | ||/ 2/x\\ | || 4 |
|||1 + cot |-|| | ||/ 2/x pi\\ | |||1 + cot |-|| | ||/ 2/x pi\\ |
||\ \2// | |||1 + cot |- + --|| | ||\ \2// | |||1 + cot |- + --|| |
\\ / ||\ \2 4 // | \\ / ||\ \2 4 // |
\\ / \\ /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{16 \cot^{4}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{16 \cot^{4}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{4}} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ // 0 for x mod pi = 0\ || |
|| | || | || | || 4 |
|| 2/x\ | || 2 | || 8/x\ | || / 1 \ 8/x\ |
|| 4*tan |-| | ||/ 2/x\\ | || 4096*tan |-| | ||256*|-1 + -------| *tan |-| |
|| \2/ | |||1 - tan |-|| | || \4/ | || | 2/x\| \4/ |
2*|<-------------- otherwise |*|<\ \2// | + |<---------------------- otherwise | + |< | tan |-|| |
|| 2 | ||-------------- otherwise | || 8 | || \ \2// |
||/ 2/x\\ | || 2 | ||/ 2/x\\ 4/x\ | ||--------------------------- otherwise |
|||1 + tan |-|| | ||/ 2/x\\ | |||1 + tan |-|| *tan |-| | || 8 |
||\ \2// | |||1 + tan |-|| | ||\ \4// \2/ | || / 2/x\\ |
\\ / \\\ \2// / \\ / || |1 + tan |-|| |
\\ \ \4// /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4096 \tan^{8}{\left(\frac{x}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{8} \tan^{4}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{256 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{4} \tan^{8}{\left(\frac{x}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{8}} & \text{otherwise} \end{cases}\right)$$
// / 3*pi\ \ // / 3*pi\ \
// 1 for x mod 2*pi = 0\ || 1 for |x + ----| mod 2*pi = 0| // 1 for x mod 2*pi = 0\ || 1 for |x + ----| mod 2*pi = 0|
|| | || \ 2 / | || | || \ 2 / |
|| 2 | || | || 4 | || |
||/ 2/x\\ | || 2 | ||/ 2/x\\ | || 4 |
|||-1 + cot |-|| | ||/ 2/x pi\\ | |||-1 + cot |-|| | ||/ 2/x pi\\ |
2*|<\ \2// |*|<|-1 + tan |- + --|| | + |<\ \2// | + |<|-1 + tan |- + --|| |
||--------------- otherwise | ||\ \2 4 // | ||--------------- otherwise | ||\ \2 4 // |
|| 2 | ||-------------------- otherwise | || 4 | ||-------------------- otherwise |
|| / 2/x\\ | || 2 | || / 2/x\\ | || 4 |
|| |1 + cot |-|| | ||/ 2/x pi\\ | || |1 + cot |-|| | ||/ 2/x pi\\ |
\\ \ \2// / |||1 + tan |- + --|| | \\ \ \2// / |||1 + tan |- + --|| |
\\\ \2 4 // / \\\ \2 4 // /
$$\left(2 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{4}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)^{4}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{4}} & \text{otherwise} \end{cases}\right)$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ || | || |
|| | || | || // x \ | || // x \ |
||/ 0 for x mod pi = 0 | ||/ 1 for x mod 2*pi = 0 | || || 0 for - mod pi = 0| | || 4 || 0 for - mod pi = 0| |
2*|<| |*|<| | + |< 4/x\ || 2 | | + |
$$\left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\16 \left(\begin{cases} 0 & \text{for}\: \frac{x}{2} \bmod \pi = 0 \\256 \sin^{16}{\left(\frac{x}{4} \right)} \cot^{8}{\left(\frac{x}{4} \right)} & \text{otherwise} \end{cases}\right) \cot^{4}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{4} \left(\begin{cases} 0 & \text{for}\: \frac{x}{2} \bmod \pi = 0 \\256 \sin^{16}{\left(\frac{x}{4} \right)} \cot^{8}{\left(\frac{x}{4} \right)} & \text{otherwise} \end{cases}\right) & \text{otherwise} \end{cases}\right)$$
// 0 for x mod pi = 0\ // 0 for x mod pi = 0\
|| | // 1 for x mod 2*pi = 0\ || | // 1 for x mod 2*pi = 0\
|| 2 | || | || 4 | || |
|| sin (x) | || 2 | || sin (x) | || 4 |
||------------------------ otherwise | ||/ 2 4/x\\ | ||------------------------ otherwise | ||/ 2 4/x\\ |
|| 2 | |||sin (x) - 4*sin |-|| | || 4 | |||sin (x) - 4*sin |-|| |
2*|
$$\left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{\sin^{2}{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(- 4 \sin^{4}{\left(\frac{x}{2} \right)} + \sin^{2}{\left(x \right)}\right)^{2}}{\left(4 \sin^{4}{\left(\frac{x}{2} \right)} + \sin^{2}{\left(x \right)}\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{\sin^{4}{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{4} \sin^{8}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(- 4 \sin^{4}{\left(\frac{x}{2} \right)} + \sin^{2}{\left(x \right)}\right)^{4}}{\left(4 \sin^{4}{\left(\frac{x}{2} \right)} + \sin^{2}{\left(x \right)}\right)^{4}} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\ // 1 for x mod 2*pi = 0\
|| | || |
// 0 for x mod pi = 0\ || 2 | // 0 for x mod pi = 0\ || 4 |
|| | ||/ 2 \ | || | ||/ 2 \ |
|| 2 | ||| sin (x) | | || 4 | ||| sin (x) | |
|| sin (x) | |||-1 + ---------| | || sin (x) | |||-1 + ---------| |
||------------------------ otherwise | ||| 4/x\| | ||------------------------ otherwise | ||| 4/x\| |
|| 2 | ||| 4*sin |-|| | || 4 | ||| 4*sin |-|| |
2*|
$$\left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{\sin^{2}{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{\sin^{4}{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{4} \sin^{8}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{4}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{4}} & \text{otherwise} \end{cases}\right)$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ || | || |
|| | || | || // x \ | || // x \ |
||/ 0 for x mod pi = 0 | ||/ 1 for x mod 2*pi = 0 | || || 0 for - mod pi = 0| | || || 0 for - mod pi = 0| |
||| | ||| | || || 2 | | || || 2 | |
||| 2/x\ | ||| 2 | || || | | || || | |
||| 4*cot |-| | |||/ 2/x\\ | || || 8/x\ | | || 4 || 8/x\ | |
2*|<| \2/ |*|<||-1 + cot |-|| | + |< 4/x\ || 256*cot |-| | | + |
$$\left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\16 \left(\begin{cases} 0 & \text{for}\: \frac{x}{2} \bmod \pi = 0 \\\frac{256 \cot^{8}{\left(\frac{x}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{8}} & \text{otherwise} \end{cases}\right) \cot^{4}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{4} \left(\begin{cases} 0 & \text{for}\: \frac{x}{2} \bmod \pi = 0 \\\frac{256 \cot^{8}{\left(\frac{x}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{8}} & \text{otherwise} \end{cases}\right) & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\ // 1 for x mod 2*pi = 0\
|| | || |
// 0 for x mod pi = 0\ || 2 | // 0 for x mod pi = 0\ || 4 |
|| | ||/ 2/x\ \ | || | ||/ 2/x\ \ |
|| 2/x\ | ||| cos |-| | | || 4/x\ | ||| cos |-| | |
|| 4*cos |-| | ||| \2/ | | || 16*cos |-| | ||| \2/ | |
|| \2/ | |||-1 + ------------| | || \2/ | |||-1 + ------------| |
||-------------------------------- otherwise | ||| 2/x pi\| | ||-------------------------------- otherwise | ||| 2/x pi\| |
|| 2 | ||| cos |- - --|| | || 4 | ||| cos |- - --|| |
2*|
$$\left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \cos^{2}{\left(\frac{x}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{16 \cos^{4}{\left(\frac{x}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{4} \cos^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1\right)^{4}}{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{4}} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\ // 1 for x mod 2*pi = 0\
|| | || |
// 0 for x mod pi = 0\ || 2 | // 0 for x mod pi = 0\ || 4 |
|| | ||/ 2/x pi\\ | || | ||/ 2/x pi\\ |
|| 2/x pi\ | ||| sec |- - --|| | || 4/x pi\ | ||| sec |- - --|| |
|| 4*sec |- - --| | ||| \2 2 /| | || 16*sec |- - --| | ||| \2 2 /| |
|| \2 2 / | |||-1 + ------------| | || \2 2 / | |||-1 + ------------| |
||--------------------------- otherwise | ||| 2/x\ | | ||--------------------------- otherwise | ||| 2/x\ | |
|| 2 | ||| sec |-| | | || 4 | ||| sec |-| | |
2*|
$$\left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \sec^{2}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{16 \sec^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{4} \sec^{4}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{4}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{4}} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\ // 1 for x mod 2*pi = 0\
|| | || |
// 0 for x mod pi = 0\ || 2 | // 0 for x mod pi = 0\ || 4 |
|| | ||/ 2/x\ \ | || | ||/ 2/x\ \ |
|| 2/x\ | ||| csc |-| | | || 4/x\ | ||| csc |-| | |
|| 4*csc |-| | ||| \2/ | | || 16*csc |-| | ||| \2/ | |
|| \2/ | |||-1 + ------------| | || \2/ | |||-1 + ------------| |
||-------------------------------- otherwise | ||| 2/pi x\| | ||-------------------------------- otherwise | ||| 2/pi x\| |
|| 2 | ||| csc |-- - -|| | || 4 | ||| csc |-- - -|| |
2*|
$$\left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \csc^{2}{\left(\frac{x}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2} \csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{16 \csc^{4}{\left(\frac{x}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1\right)^{4} \csc^{4}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - 1\right)^{4}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1\right)^{4}} & \text{otherwise} \end{cases}\right)$$
2*Piecewise((0, Mod(x = pi, 0)), (4*csc(x/2)^2/((1 + csc(x/2)^2/csc(pi/2 - x/2)^2)^2*csc(pi/2 - x/2)^2), True))*Piecewise((1, Mod(x = 2*pi, 0)), ((-1 + csc(x/2)^2/csc(pi/2 - x/2)^2)^2/(1 + csc(x/2)^2/csc(pi/2 - x/2)^2)^2, True)) + Piecewise((0, Mod(x = pi, 0)), (16*csc(x/2)^4/((1 + csc(x/2)^2/csc(pi/2 - x/2)^2)^4*csc(pi/2 - x/2)^4), True)) + Piecewise((1, Mod(x = 2*pi, 0)), ((-1 + csc(x/2)^2/csc(pi/2 - x/2)^2)^4/(1 + csc(x/2)^2/csc(pi/2 - x/2)^2)^4, True))