Господин Экзамен
sin(x)^4-cos(x)^2*sin(x)^2+cos(x)^4-sin(x)^6 если x=3
Решение
Вы ввели
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4 2 2 4 6 sin (x) - cos (x)*sin (x) + cos (x) - sin (x)
$$- \sin^{6}{\left(x \right)} + \sin^{4}{\left(x \right)} - \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} + \cos^{4}{\left(x \right)}$$
sin(x)^4 - cos(x)^2*sin(x)^2 + cos(x)^4 - sin(x)^6
Подстановка условия
[src]
sin(x)^4 - cos(x)^2*sin(x)^2 + cos(x)^4 - sin(x)^6 при x = 3
подставляем
4 2 2 4 6 sin (x) - cos (x)*sin (x) + cos (x) - sin (x)
$$- \sin^{6}{\left(x \right)} + \sin^{4}{\left(x \right)} - \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} + \cos^{4}{\left(x \right)}$$
6 cos (x)
$$\cos^{6}{\left(x \right)}$$
переменные
x = 3
$$x = 3$$
6 cos ((3))
$$\cos^{6}{\left((3) \right)}$$
6 cos (3)
$$\cos^{6}{\left(3 \right)}$$
cos(3)^6
Собрать выражение
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5 cos(6*x) 3*cos(4*x) 15*cos(2*x) -- + -------- + ---------- + ----------- 16 32 16 32
$$\frac{15 \cos{\left(2 x \right)}}{32} + \frac{3 \cos{\left(4 x \right)}}{16} + \frac{\cos{\left(6 x \right)}}{32} + \frac{5}{16}$$
5/16 + cos(6*x)/32 + 3*cos(4*x)/16 + 15*cos(2*x)/32
Степени
[src]
2
/ I*x -I*x\ 2
4 4 6 |e e | / -I*x I*x\
/ I*x -I*x\ / -I*x I*x\ / -I*x I*x\ |---- + -----| *\- e + e /
|e e | \- e + e / \- e + e / \ 2 2 /
|---- + -----| + ----------------- + ----------------- + ---------------------------------
\ 2 2 / 16 64 4
$$\frac{\left(e^{i x} - e^{- i x}\right)^{6}}{64} + \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{4} + \frac{\left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{2} \left(e^{i x} - e^{- i x}\right)^{2}}{4} + \frac{\left(e^{i x} - e^{- i x}\right)^{4}}{16}$$
(exp(i*x)/2 + exp(-i*x)/2)^4 + (-exp(-i*x) + exp(i*x))^4/16 + (-exp(-i*x) + exp(i*x))^6/64 + (exp(i*x)/2 + exp(-i*x)/2)^2*(-exp(-i*x) + exp(i*x))^2/4
Тригонометрическая часть
[src]
6 cos (x)
$$\cos^{6}{\left(x \right)}$$
1 ------- 6 sec (x)
$$\frac{1}{\sec^{6}{\left(x \right)}}$$
6/ pi\
sin |x + --|
\ 2 /
$$\sin^{6}{\left(x + \frac{\pi}{2} \right)}$$
1
------------
6/pi \
csc |-- - x|
\2 /
$$\frac{1}{\csc^{6}{\left(- x + \frac{\pi}{2} \right)}}$$
6 / 2/x\\ 12/x\ |1 - tan |-|| *cos |-| \ \2// \2/
$$\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6} \cos^{12}{\left(\frac{x}{2} \right)}$$
6
/ 2/x\\
|1 - tan |-||
\ \2//
--------------
6
/ 2/x\\
|1 + tan |-||
\ \2//
$$\frac{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}}$$
4 4/ pi\ 6 2 2/ pi\
sin (x) + sin |x + --| - sin (x) - sin (x)*sin |x + --|
\ 2 / \ 2 /
$$- \sin^{6}{\left(x \right)} + \sin^{4}{\left(x \right)} - \sin^{2}{\left(x \right)} \sin^{2}{\left(x + \frac{\pi}{2} \right)} + \sin^{4}{\left(x + \frac{\pi}{2} \right)}$$
1 1 1 1 ------- + ------- - ------- - --------------- 4 4 6 2 2 csc (x) sec (x) csc (x) csc (x)*sec (x)
$$\frac{1}{\sec^{4}{\left(x \right)}} - \frac{1}{\csc^{2}{\left(x \right)} \sec^{2}{\left(x \right)}} + \frac{1}{\csc^{4}{\left(x \right)}} - \frac{1}{\csc^{6}{\left(x \right)}}$$
/ 1 for x mod 2*pi = 0
|
| 6
$$\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{6} \sin^{12}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}$$
4 4/ pi\ 6/ pi\ 2 2/ pi\
cos (x) + cos |x - --| - cos |x - --| - cos (x)*cos |x - --|
\ 2 / \ 2 / \ 2 /
$$- \cos^{6}{\left(x - \frac{\pi}{2} \right)} + \cos^{4}{\left(x \right)} - \cos^{2}{\left(x \right)} \cos^{2}{\left(x - \frac{\pi}{2} \right)} + \cos^{4}{\left(x - \frac{\pi}{2} \right)}$$
/ 1 for x mod 2*pi = 0 | | 6 |/ 2/x\\ ||-1 + cot |-|| <\ \2// |--------------- otherwise | 6 | / 2/x\\ | |1 + cot |-|| \ \ \2//
$$\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{6}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases}$$
1 1 1 1
------- + ------------ - ------- - --------------------
4 4/pi \ 6 2 2/pi \
csc (x) csc |-- - x| csc (x) csc (x)*csc |-- - x|
\2 / \2 /
$$\frac{1}{\csc^{4}{\left(- x + \frac{\pi}{2} \right)}} - \frac{1}{\csc^{2}{\left(x \right)} \csc^{2}{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{\csc^{4}{\left(x \right)}} - \frac{1}{\csc^{6}{\left(x \right)}}$$
1 1 1 1
------- + ------------ - ------------ - --------------------
4 4/ pi\ 6/ pi\ 2 2/ pi\
sec (x) sec |x - --| sec |x - --| sec (x)*sec |x - --|
\ 2 / \ 2 / \ 2 /
$$\frac{1}{\sec^{4}{\left(x - \frac{\pi}{2} \right)}} - \frac{1}{\sec^{2}{\left(x \right)} \sec^{2}{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{\sec^{4}{\left(x \right)}} - \frac{1}{\sec^{6}{\left(x - \frac{\pi}{2} \right)}}$$
1 1 1 1
------- + ------------ - ------------ - --------------------
4 4/pi \ 6/pi \ 2 2/pi \
sec (x) sec |-- - x| sec |-- - x| sec (x)*sec |-- - x|
\2 / \2 / \2 /
$$\frac{1}{\sec^{4}{\left(- x + \frac{\pi}{2} \right)}} - \frac{1}{\sec^{2}{\left(x \right)} \sec^{2}{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{\sec^{4}{\left(x \right)}} - \frac{1}{\sec^{6}{\left(- x + \frac{\pi}{2} \right)}}$$
1 1 1 1
------------ + ------------ - ------------ - -------------------------
4 4/pi \ 6 2 2/pi \
csc (pi - x) csc |-- - x| csc (pi - x) csc (pi - x)*csc |-- - x|
\2 / \2 /
$$\frac{1}{\csc^{4}{\left(- x + \frac{\pi}{2} \right)}} - \frac{1}{\csc^{2}{\left(- x + \pi \right)} \csc^{2}{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{\csc^{4}{\left(- x + \pi \right)}} - \frac{1}{\csc^{6}{\left(- x + \pi \right)}}$$
6 6
(1 - cos(x) + sin(x)) *(-1 + cos(x) + sin(x))
----------------------------------------------
6
/1 2 cos(2*x)\
|- + (1 - cos(x)) - --------|
\2 2 /
$$\frac{\left(\sin{\left(x \right)} - \cos{\left(x \right)} + 1\right)^{6} \left(\sin{\left(x \right)} + \cos{\left(x \right)} - 1\right)^{6}}{\left(\left(- \cos{\left(x \right)} + 1\right)^{2} - \frac{\cos{\left(2 x \right)}}{2} + \frac{1}{2}\right)^{6}}$$
7 cos(4*x) cos(6*x) 15*cos(2*x) /1 cos(2*x)\ /1 cos(2*x)\ -- + -------- + -------- + ----------- - |- + --------|*|- - --------| 16 16 32 32 \2 2 / \2 2 /
$$- \left(- \frac{\cos{\left(2 x \right)}}{2} + \frac{1}{2}\right) \left(\frac{\cos{\left(2 x \right)}}{2} + \frac{1}{2}\right) + \frac{15 \cos{\left(2 x \right)}}{32} + \frac{\cos{\left(4 x \right)}}{16} + \frac{\cos{\left(6 x \right)}}{32} + \frac{7}{16}$$
6/x\ 6/x\ 8/x\ 4/x\ 8/x\ 6/x\ 6/x\
- 68*sin |-| - 8*cos |-| + 9*cos |-| + 30*sin |-| + 39*sin |-| - 64*cos |-|*sin |-|
\2/ \2/ \2/ \2/ \2/ \2/ \2/
$$- 64 \sin^{6}{\left(\frac{x}{2} \right)} \cos^{6}{\left(\frac{x}{2} \right)} + 39 \sin^{8}{\left(\frac{x}{2} \right)} + 9 \cos^{8}{\left(\frac{x}{2} \right)} - 68 \sin^{6}{\left(\frac{x}{2} \right)} - 8 \cos^{6}{\left(\frac{x}{2} \right)} + 30 \sin^{4}{\left(\frac{x}{2} \right)}$$
4 4 4/x\ 6 6/x\ 2 2/x\
cos (x) + (1 + cos(x)) *tan |-| - (1 + cos(x)) *tan |-| - 2*cos (x)*sin |-|*(1 + cos(x))
\2/ \2/ \2/
$$- \left(\cos{\left(x \right)} + 1\right)^{6} \tan^{6}{\left(\frac{x}{2} \right)} + \left(\cos{\left(x \right)} + 1\right)^{4} \tan^{4}{\left(\frac{x}{2} \right)} - 2 \left(\cos{\left(x \right)} + 1\right) \sin^{2}{\left(\frac{x}{2} \right)} \cos^{2}{\left(x \right)} + \cos^{4}{\left(x \right)}$$
6/x\ 6/pi x\ 8/pi x\ 4/x\ 8/x\ 6/x\ 6/pi x\
- 68*sin |-| - 8*sin |-- + -| + 9*sin |-- + -| + 30*sin |-| + 39*sin |-| - 64*sin |-|*sin |-- + -|
\2/ \2 2/ \2 2/ \2/ \2/ \2/ \2 2/
$$- 64 \sin^{6}{\left(\frac{x}{2} \right)} \sin^{6}{\left(\frac{x}{2} + \frac{\pi}{2} \right)} + 39 \sin^{8}{\left(\frac{x}{2} \right)} + 9 \sin^{8}{\left(\frac{x}{2} + \frac{\pi}{2} \right)} - 68 \sin^{6}{\left(\frac{x}{2} \right)} - 8 \sin^{6}{\left(\frac{x}{2} + \frac{\pi}{2} \right)} + 30 \sin^{4}{\left(\frac{x}{2} \right)}$$
6/x pi\ 6/x\ 8/x\ 4/x pi\ 8/x pi\ 6/x\ 6/x pi\
- 68*cos |- - --| - 8*cos |-| + 9*cos |-| + 30*cos |- - --| + 39*cos |- - --| - 64*cos |-|*cos |- - --|
\2 2 / \2/ \2/ \2 2 / \2 2 / \2/ \2 2 /
$$- 64 \cos^{6}{\left(\frac{x}{2} \right)} \cos^{6}{\left(\frac{x}{2} - \frac{\pi}{2} \right)} + 9 \cos^{8}{\left(\frac{x}{2} \right)} + 39 \cos^{8}{\left(\frac{x}{2} - \frac{\pi}{2} \right)} - 8 \cos^{6}{\left(\frac{x}{2} \right)} - 68 \cos^{6}{\left(\frac{x}{2} - \frac{\pi}{2} \right)} + 30 \cos^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}$$
68 8 9 30 39 64
- ------- - ------------ + ------------ + ------- + ------- - --------------------
6/x\ 6/pi x\ 8/pi x\ 4/x\ 8/x\ 6/x\ 6/pi x\
csc |-| csc |-- - -| csc |-- - -| csc |-| csc |-| csc |-|*csc |-- - -|
\2/ \2 2/ \2 2/ \2/ \2/ \2/ \2 2/
$$\frac{30}{\csc^{4}{\left(\frac{x}{2} \right)}} - \frac{8}{\csc^{6}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - \frac{68}{\csc^{6}{\left(\frac{x}{2} \right)}} + \frac{9}{\csc^{8}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + \frac{39}{\csc^{8}{\left(\frac{x}{2} \right)}} - \frac{64}{\csc^{6}{\left(\frac{x}{2} \right)} \csc^{6}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}$$
68 8 9 30 39 64
- ------------ - ------- + ------- + ------------ + ------------ - --------------------
6/x pi\ 6/x\ 8/x\ 4/x pi\ 8/x pi\ 6/x\ 6/x pi\
sec |- - --| sec |-| sec |-| sec |- - --| sec |- - --| sec |-|*sec |- - --|
\2 2 / \2/ \2/ \2 2 / \2 2 / \2/ \2 2 /
$$\frac{30}{\sec^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - \frac{68}{\sec^{6}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - \frac{8}{\sec^{6}{\left(\frac{x}{2} \right)}} + \frac{39}{\sec^{8}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + \frac{9}{\sec^{8}{\left(\frac{x}{2} \right)}} - \frac{64}{\sec^{6}{\left(\frac{x}{2} \right)} \sec^{6}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}$$
2 2
4/x pi\ / /x\\ / /x\\ 4/x\
16*tan |- + --| |1 + tan|-|| *|-1 + tan|-|| *cos |-|*(1 - cos(2*x))
12/x\ 6/x\ \2 4 / 8/x\ 4/x\ \ \2// \ \2// \2/
- 64*cos |-|*tan |-| + ------------------- + 16*cos |-|*tan |-| - ---------------------------------------------------
\2/ \2/ 4 \2/ \2/ 2
/ 2/x pi\\
|1 + tan |- + --||
\ \2 4 //
$$- 64 \cos^{12}{\left(\frac{x}{2} \right)} \tan^{6}{\left(\frac{x}{2} \right)} + 16 \cos^{8}{\left(\frac{x}{2} \right)} \tan^{4}{\left(\frac{x}{2} \right)} - \frac{\left(- \cos{\left(2 x \right)} + 1\right) \left(\tan{\left(\frac{x}{2} \right)} - 1\right)^{2} \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2} \cos^{4}{\left(\frac{x}{2} \right)}}{2} + \frac{16 \tan^{4}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{4}}$$
2 2
4/x pi\ / /x\\ / /x\\ 4/x\
16*tan |- + --| |1 + tan|-|| *|-1 + tan|-|| *cos |-|*(1 - cos(2*x))
6/x\ 12/x\ \2 4 / 4/x\ 8/x\ \ \2// \ \2// \2/
- 64*cot |-|*sin |-| + ------------------- + 16*cot |-|*sin |-| - ---------------------------------------------------
\2/ \2/ 4 \2/ \2/ 2
/ 2/x pi\\
|1 + tan |- + --||
\ \2 4 //
$$- 64 \sin^{12}{\left(\frac{x}{2} \right)} \cot^{6}{\left(\frac{x}{2} \right)} + 16 \sin^{8}{\left(\frac{x}{2} \right)} \cot^{4}{\left(\frac{x}{2} \right)} - \frac{\left(- \cos{\left(2 x \right)} + 1\right) \left(\tan{\left(\frac{x}{2} \right)} - 1\right)^{2} \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2} \cos^{4}{\left(\frac{x}{2} \right)}}{2} + \frac{16 \tan^{4}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{4}}$$
4 2
/ 2/x\\ 6/x\ 4/x\ / 2/x\\ 2/x\
|1 - tan |-|| 64*tan |-| 16*tan |-| 4*|1 - tan |-|| *tan |-|
\ \2// \2/ \2/ \ \2// \2/
-------------- - -------------- + -------------- - ------------------------
4 6 4 4
/ 2/x\\ / 2/x\\ / 2/x\\ / 2/x\\
|1 + tan |-|| |1 + tan |-|| |1 + tan |-|| |1 + tan |-||
\ \2// \ \2// \ \2// \ \2//
$$\frac{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}} - \frac{4 \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}} + \frac{16 \tan^{4}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}} - \frac{64 \tan^{6}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}}$$
2 2
4/x\ / 2/x\\ / 2/x pi\\ 4/x\
4 16*tan |-| |-1 + cot |-|| *|-1 + tan |- + --|| *sin |-|
/ 2/x\\ 8/x\ 6/x\ 6/x\ \2/ \ \2// \ \2 4 // \2/
|-1 + cot |-|| *sin |-| - 64*cos |-|*sin |-| + -------------- - --------------------------------------------
\ \2// \2/ \2/ \2/ 4 2
/ 2/x\\ / 2/x pi\\
|1 + tan |-|| |1 + tan |- + --||
\ \2// \ \2 4 //
$$\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{4} \sin^{8}{\left(\frac{x}{2} \right)} - 64 \sin^{6}{\left(\frac{x}{2} \right)} \cos^{6}{\left(\frac{x}{2} \right)} - \frac{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2} \sin^{4}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} + \frac{16 \tan^{4}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}}$$
2
2 2 / 2 2 \ 2 2 / 2 2 \ / 2 2 \ 2 2
7 15*sin (x) sin (3*x) \cos (x) - sin (x)/ cos (3*x) 15*cos (x) |1 cos (x) sin (x)| |1 sin (x) cos (x)| cos (x)*sin (x)
-- - ---------- - --------- + -------------------- + --------- + ---------- - |- + ------- - -------|*|- + ------- - -------| - ---------------
16 32 32 16 32 32 \2 2 2 / \2 2 2 / 4
$$- \frac{\sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{4} + \frac{\left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)^{2}}{16} - \left(- \frac{\sin^{2}{\left(x \right)}}{2} + \frac{\cos^{2}{\left(x \right)}}{2} + \frac{1}{2}\right) \left(\frac{\sin^{2}{\left(x \right)}}{2} - \frac{\cos^{2}{\left(x \right)}}{2} + \frac{1}{2}\right) - \frac{15 \sin^{2}{\left(x \right)}}{32} - \frac{\sin^{2}{\left(3 x \right)}}{32} + \frac{15 \cos^{2}{\left(x \right)}}{32} + \frac{\cos^{2}{\left(3 x \right)}}{32} + \frac{7}{16}$$
4 2
/ 1 \ / 1 \
|1 - -------| 4*|1 - -------|
| 2/x\| | 2/x\|
| cot |-|| | cot |-||
\ \2// 64 16 \ \2//
-------------- - ---------------------- + ---------------------- - ----------------------
4 6 4 4
/ 1 \ / 1 \ 6/x\ / 1 \ 4/x\ / 1 \ 2/x\
|1 + -------| |1 + -------| *cot |-| |1 + -------| *cot |-| |1 + -------| *cot |-|
| 2/x\| | 2/x\| \2/ | 2/x\| \2/ | 2/x\| \2/
| cot |-|| | cot |-|| | cot |-|| | cot |-||
\ \2// \ \2// \ \2// \ \2//
$$\frac{\left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{4}}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{4}} - \frac{4 \left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{4} \cot^{2}{\left(\frac{x}{2} \right)}} + \frac{16}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{4} \cot^{4}{\left(\frac{x}{2} \right)}} - \frac{64}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{6} \cot^{6}{\left(\frac{x}{2} \right)}}$$
6 8 6
12/x\ 6/x\ / 2/x\\ 12/x\ / 2/x\\ 16/x\ 8/x\ 4/x\ 16/x\ 8/x\ / 2/x\\ 24/x\ 6/x\
- 4352*cos |-|*tan |-| - 8*|1 - tan |-|| *cos |-| + 9*|1 - tan |-|| *cos |-| + 480*cos |-|*tan |-| + 9984*cos |-|*tan |-| - 4096*|1 - tan |-|| *cos |-|*tan |-|
\4/ \4/ \ \4// \4/ \ \4// \4/ \4/ \4/ \4/ \4/ \ \4// \4/ \4/
$$- 4096 \left(- \tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{6} \cos^{24}{\left(\frac{x}{4} \right)} \tan^{6}{\left(\frac{x}{4} \right)} + 9 \left(- \tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{8} \cos^{16}{\left(\frac{x}{4} \right)} + 9984 \cos^{16}{\left(\frac{x}{4} \right)} \tan^{8}{\left(\frac{x}{4} \right)} - 8 \left(- \tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{6} \cos^{12}{\left(\frac{x}{4} \right)} - 4352 \cos^{12}{\left(\frac{x}{4} \right)} \tan^{6}{\left(\frac{x}{4} \right)} + 480 \cos^{8}{\left(\frac{x}{4} \right)} \tan^{4}{\left(\frac{x}{4} \right)}$$
6/x\ 4/x\ 4/x pi\ 2/x\ 2/x pi\
64*tan |-| 16*tan |-| 16*tan |- + --| 16*tan |-|*tan |- + --|
\2/ \2/ \2 4 / \2/ \2 4 /
- -------------- + -------------- + ------------------- - ----------------------------------
6 4 4 2 2
/ 2/x\\ / 2/x\\ / 2/x pi\\ / 2/x\\ / 2/x pi\\
|1 + tan |-|| |1 + tan |-|| |1 + tan |- + --|| |1 + tan |-|| *|1 + tan |- + --||
\ \2// \ \2// \ \2 4 // \ \2// \ \2 4 //
$$\frac{16 \tan^{4}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{4}} - \frac{16 \tan^{2}{\left(\frac{x}{2} \right)} \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} + \frac{16 \tan^{4}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}} - \frac{64 \tan^{6}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}}$$
6/x\ 4/x\ 4/x pi\ 2/x\ 2/x pi\
64*cot |-| 16*cot |-| 16*tan |- + --| 16*cot |-|*tan |- + --|
\2/ \2/ \2 4 / \2/ \2 4 /
- -------------- + -------------- + ------------------- - ----------------------------------
6 4 4 2 2
/ 2/x\\ / 2/x\\ / 2/x pi\\ / 2/x\\ / 2/x pi\\
|1 + cot |-|| |1 + cot |-|| |1 + tan |- + --|| |1 + cot |-|| *|1 + tan |- + --||
\ \2// \ \2// \ \2 4 // \ \2// \ \2 4 //
$$\frac{16 \cot^{4}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}} - \frac{64 \cot^{6}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}} - \frac{16 \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + \frac{16 \tan^{4}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{4}}$$
6 4 2 2
/ 2/x pi\\ 6 / 2/x pi\\ 4 / 2/x pi\\ / 2/x\\ 2 4/x\
4 |1 - cot |- + --|| *(1 + sin(x)) |1 - cot |- + --|| *(1 + sin(x)) |1 - cot |- + --|| *|1 - tan |-|| *(1 + sin(x)) *cos |-|
/ 2/x\\ 8/x\ \ \2 4 // \ \2 4 // \ \2 4 // \ \2// \2/
|1 - tan |-|| *cos |-| - --------------------------------- + --------------------------------- - --------------------------------------------------------
\ \2// \2/ 64 16 4
$$\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4} \cos^{8}{\left(\frac{x}{2} \right)} - \frac{\left(- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{6} \left(\sin{\left(x \right)} + 1\right)^{6}}{64} - \frac{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \left(- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\sin{\left(x \right)} + 1\right)^{2} \cos^{4}{\left(\frac{x}{2} \right)}}{4} + \frac{\left(- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{4} \left(\sin{\left(x \right)} + 1\right)^{4}}{16}$$
4 4 6 2 2
/ 2/x\\ / 2/x pi\\ / 2/x pi\\ / 2/x\\ / 2/x pi\\
|-1 + cot |-|| |-1 + tan |- + --|| |-1 + tan |- + --|| |-1 + cot |-|| *|-1 + tan |- + --||
\ \2// \ \2 4 // \ \2 4 // \ \2// \ \2 4 //
--------------- + -------------------- - -------------------- - ------------------------------------
4 4 6 2 2
/ 2/x\\ / 2/x pi\\ / 2/x pi\\ / 2/x\\ / 2/x pi\\
|1 + cot |-|| |1 + tan |- + --|| |1 + tan |- + --|| |1 + cot |-|| *|1 + tan |- + --||
\ \2// \ \2 4 // \ \2 4 // \ \2// \ \2 4 //
$$- \frac{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)^{6}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{6}} + \frac{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)^{4}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{4}} - \frac{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + \frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{4}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}}$$
4 4 6 2 2
/ 2/x pi\\ / 2/x\\ / 2/x pi\\ / 2/x pi\\ / 2/x\\
|1 - cot |- + --|| |1 - tan |-|| |1 - cot |- + --|| |1 - cot |- + --|| *|1 - tan |-||
\ \2 4 // \ \2// \ \2 4 // \ \2 4 // \ \2//
------------------- + -------------- - ------------------- - ----------------------------------
4 4 6 2 2
/ 2/x pi\\ / 2/x\\ / 2/x pi\\ / 2/x pi\\ / 2/x\\
|1 + cot |- + --|| |1 + tan |-|| |1 + cot |- + --|| |1 + cot |- + --|| *|1 + tan |-||
\ \2 4 // \ \2// \ \2 4 // \ \2 4 // \ \2//
$$\frac{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}} - \frac{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \left(- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} - \frac{\left(- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{6}}{\left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{6}} + \frac{\left(- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{4}}{\left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{4}}$$
// 1 for x mod 2*pi = 0\
|| |
// 0 for x mod pi = 0\ // 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ // 0 for x mod pi = 0\ || 4 |
|| | || | || | || | ||/ 2 \ |
- |< 6 | - |< 2 |*|< 2 | + |< 4 | + |<| sin (x) | 8/x\ |
||sin (x) otherwise | ||sin (x) otherwise | ||cos (x) otherwise | ||sin (x) otherwise | |||-1 + ---------| *sin |-| otherwise |
\\ / \\ / \\ / \\ / ||| 4/x\| \2/ |
||| 4*sin |-|| |
\\\ \2// /
$$\left(- \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{4}{\left(x \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{6}{\left(x \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\left(-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{4} \sin^{8}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\
// 0 for x mod pi = 0\ // 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ // 0 for x mod pi = 0\ || |
|| | || | || | || | || 4 |
- |< 6 | - |< 2 |*|< 2 | + |< 4/x\ 8/x\ | + |
$$\left(- \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\16 \sin^{8}{\left(\frac{x}{2} \right)} \cot^{4}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{6}{\left(x \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{4} \sin^{8}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\
// 0 for x mod pi = 0\ // 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ // 0 for x mod pi = 0\ || |
|| | || | || | || | || 4 |
- |< 6 | - |< 2 |*|< 2 | + |< 4/x\ 4/x\ | + |
$$\left(- \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\16 \sin^{4}{\left(\frac{x}{2} \right)} \cos^{4}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{6}{\left(x \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4} \cos^{8}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\
|| |
// 0 for x mod pi = 0\ // 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ // 0 for x mod pi = 0\ || 4 |
|| | || | || | || | ||/ 2 \ |
- |< 6 | - |< 2 |*|< 2/ pi\ | + |< 4 | + |<| sin (x) | 8/x\ |
||sin (x) otherwise | ||sin (x) otherwise | ||sin |x + --| otherwise | ||sin (x) otherwise | |||-1 + ---------| *sin |-| otherwise |
\\ / \\ / \\ \ 2 / / \\ / ||| 4/x\| \2/ |
||| 4*sin |-|| |
\\\ \2// /
$$\left(- \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\sin^{2}{\left(x + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{4}{\left(x \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{6}{\left(x \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\left(-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{4} \sin^{8}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right)$$
// 0 for x mod pi = 0\ // 0 for x mod pi = 0\ || | || | || 6 | // 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ || 4 | // 1 for x mod 2*pi = 0\ ||(-1 + cos(x)) | || | || | ||(-1 + cos(x)) | || | - |<-------------- otherwise | - |< 2 |*|< 2 | + |<-------------- otherwise | + |< 4 | || 6/x\ | ||sin (x) otherwise | ||cos (x) otherwise | || 4/x\ | ||cos (x) otherwise | || tan |-| | \\ / \\ / || tan |-| | \\ / || \2/ | || \2/ | \\ / \\ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{\left(\cos{\left(x \right)} - 1\right)^{4}}{\tan^{4}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{\left(\cos{\left(x \right)} - 1\right)^{6}}{\tan^{6}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{4}{\left(x \right)} & \text{otherwise} \end{cases}\right)$$
6 8 6
6/x\ / 2/x\\ / 2/x\\ 4/x\ 8/x\ / 2/x\\ 6/x\
4352*tan |-| 8*|1 - tan |-|| 9*|1 - tan |-|| 480*tan |-| 9984*tan |-| 4096*|1 - tan |-|| *tan |-|
\4/ \ \4// \ \4// \4/ \4/ \ \4// \4/
- -------------- - ---------------- + ---------------- + -------------- + -------------- - ---------------------------
6 6 8 4 8 12
/ 2/x\\ / 2/x\\ / 2/x\\ / 2/x\\ / 2/x\\ / 2/x\\
|1 + tan |-|| |1 + tan |-|| |1 + tan |-|| |1 + tan |-|| |1 + tan |-|| |1 + tan |-||
\ \4// \ \4// \ \4// \ \4// \ \4// \ \4//
$$\frac{9 \left(- \tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{8}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{8}} - \frac{8 \left(- \tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{6}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{6}} - \frac{4096 \left(- \tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{6} \tan^{6}{\left(\frac{x}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{12}} + \frac{480 \tan^{4}{\left(\frac{x}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4}} - \frac{4352 \tan^{6}{\left(\frac{x}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{6}} + \frac{9984 \tan^{8}{\left(\frac{x}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{8}}$$
// 0 for x mod pi = 0\
|| | // 1 for x mod 2*pi = 0\
// 0 for x mod pi = 0\ // 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ || 8/x\ | || |
|| | || | || | ||16*sin |-| | || 4 |
- |< 6 | - |< 2 |*|< 2 | + |< \2/ | + |
$$\left(- \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{16 \sin^{8}{\left(\frac{x}{2} \right)}}{\tan^{4}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{6}{\left(x \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4} \cos^{8}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\
|| |
// 0 for x mod pi = 0\ // 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 |-| |
- |< 6 | - |< 2 |*|< 2 | + |< 4/x\ 4/x\ | + |< \2/ |
||sin (x) otherwise | ||sin (x) otherwise | ||cos (x) otherwise | ||16*cos |-|*sin |-| otherwise | ||------------------ otherwise |
\\ / \\ / \\ / \\ \2/ \2/ / || 4 |
|| (-1 + cos(x)) |
\\ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\16 \sin^{4}{\left(\frac{x}{2} \right)} \cos^{4}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{6}{\left(x \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{16 \sin^{8}{\left(\frac{x}{2} \right)} \cos^{4}{\left(x \right)}}{\left(\cos{\left(x \right)} - 1\right)^{4}} & \text{otherwise} \end{cases}\right)$$
4 2
/ 4/x\\ / 4/x\\
| 4*sin |-|| | 4*sin |-||
| \2/| | \2/| 4/x\
|1 - ---------| 12/x\ 8/x\ 16*|1 - ---------| *sin |-|
| 2 | 4096*sin |-| 256*sin |-| | 2 | \2/
\ sin (x) / \2/ \2/ \ sin (x) /
---------------- - ------------------------ + ------------------------ - ---------------------------
4 6 4 4
/ 4/x\\ / 4/x\\ / 4/x\\ / 4/x\\
| 4*sin |-|| | 4*sin |-|| | 4*sin |-|| | 4*sin |-||
| \2/| | \2/| 6 | \2/| 4 | \2/| 2
|1 + ---------| |1 + ---------| *sin (x) |1 + ---------| *sin (x) |1 + ---------| *sin (x)
| 2 | | 2 | | 2 | | 2 |
\ sin (x) / \ sin (x) / \ sin (x) / \ sin (x) /
$$\frac{\left(- \frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{4}}{\left(\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{4}} - \frac{16 \left(- \frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{2} \sin^{4}{\left(\frac{x}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{4} \sin^{2}{\left(x \right)}} + \frac{256 \sin^{8}{\left(\frac{x}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{4} \sin^{4}{\left(x \right)}} - \frac{4096 \sin^{12}{\left(\frac{x}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{6} \sin^{6}{\left(x \right)}}$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\
|| | || |
// 0 for x mod 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\ |
- |< 6 | - |< 2 |*|< 2 | + |< \4/ \4/ | + |<256*|-1 + -------| *cos |-|*tan |-| otherwise |
||sin (x) otherwise | ||sin (x) otherwise | ||cos (x) otherwise | ||--------------------- otherwise | || | 2/x\| \4/ \4/ |
\\ / \\ / \\ / || 4/x\ | || | tan |-|| |
|| tan |-| | || \ \2// |
\\ \2/ / \\ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4096 \cos^{16}{\left(\frac{x}{4} \right)} \tan^{8}{\left(\frac{x}{4} \right)}}{\tan^{4}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{6}{\left(x \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\256 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{4} \cos^{16}{\left(\frac{x}{4} \right)} \tan^{8}{\left(\frac{x}{4} \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\
|| |
|| 4 |
// 0 for x mod pi = 0\ // 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ // 0 for x mod pi = 0\ ||/ 2/x\ \ |
|| | || | || | || | ||| cos |-| | |
- |< 6/ pi\ | - |< 2/ pi\ |*|< 2 | + |< 4/x\ 4/x pi\ | + |<| \2/ | 8/x pi\ |
||cos |x - --| otherwise | ||cos |x - --| otherwise | ||cos (x) otherwise | ||16*cos |-|*cos |- - --| otherwise | |||-1 + ------------| *cos |- - --| otherwise |
\\ \ 2 / / \\ \ 2 / / \\ / \\ \2/ \2 2 / / ||| 2/x pi\| \2 2 / |
||| cos |- - --|| |
||\ \2 2 // |
\\ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\cos^{2}{\left(x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\16 \cos^{4}{\left(\frac{x}{2} \right)} \cos^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\cos^{6}{\left(x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1\right)^{4} \cos^{8}{\left(\frac{x}{2} - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)$$
// / 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\ || 0 for |x + --| mod pi = 0|
|| | || | || \ 2 / | || | || \ 2 / |
- |< 6 | - |< 2 |*|< | + |< 4/x\ 8/x\ | + |< |
||sin (x) otherwise | ||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 / /
$$\left(- \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(x \right)} + 1\right)^{2} \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\16 \sin^{8}{\left(\frac{x}{2} \right)} \cot^{4}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{6}{\left(x \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(x \right)} + 1\right)^{4} \cot^{4}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\
|| |
|| 4 |
||/ 2/x\ \ |
// 0 for x mod pi = 0\ // 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ // 0 for x mod pi = 0\ ||| csc |-| | |
|| | || | || | || | ||| \2/ | |
|| 1 | || 1 | || 1 | || 16 | |||-1 + ------------| |
- |<------- otherwise | - |<------- otherwise |*|<------------ otherwise | + |<-------------------- otherwise | + |<| 2/pi x\| |
|| 6 | || 2 | || 2/pi \ | || 4/x\ 4/pi x\ | ||| csc |-- - -|| |
||csc (x) | ||csc (x) | ||csc |-- - x| | ||csc |-|*csc |-- - -| | ||\ \2 2// |
\\ / \\ / \\ \2 / / \\ \2/ \2 2/ / ||-------------------- otherwise |
|| 8/x\ |
|| csc |-| |
|| \2/ |
\\ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\csc^{2}{\left(x \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{1}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{16}{\csc^{4}{\left(\frac{x}{2} \right)} \csc^{4}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\csc^{6}{\left(x \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - 1\right)^{4}}{\csc^{8}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\
|| |
|| 4 |
||/ 2/x pi\\ |
// 0 for x mod pi = 0\ // 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ // 0 for x mod pi = 0\ ||| sec |- - --|| |
|| | || | || | || | ||| \2 2 /| |
|| 1 | || 1 | || 1 | || 16 | |||-1 + ------------| |
- |<------------ otherwise | - |<------------ otherwise |*|<------- otherwise | + |<-------------------- otherwise | + |<| 2/x\ | |
|| 6/ pi\ | || 2/ pi\ | || 2 | || 4/x\ 4/x pi\ | ||| sec |-| | |
||sec |x - --| | ||sec |x - --| | ||sec (x) | ||sec |-|*sec |- - --| | ||\ \2/ / |
\\ \ 2 / / \\ \ 2 / / \\ / \\ \2/ \2 2 / / ||-------------------- otherwise |
|| 8/x pi\ |
|| sec |- - --| |
|| \2 2 / |
\\ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{1}{\sec^{2}{\left(x \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{16}{\sec^{4}{\left(\frac{x}{2} \right)} \sec^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\sec^{6}{\left(x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{4}}{\sec^{8}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// / 3*pi\ \
|| 1 for |x + ----| mod 2*pi = 0|
|| \ 2 / |
// / 3*pi\ \ // / 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| || | || 4/x\ |
|| \ 2 / | || | || \ 2 / | || 4 | || 16*tan |-| |
- |< | - |< 2 |*|< | + |
$$\left(- \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\- 4 \cos^{4}{\left(\frac{x}{2} \right)} + 4 \cos^{2}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{4} \sin^{8}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{16 \tan^{4}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin^{6}{\left(x \right)} & \text{otherwise} \end{cases}\right)$$
4 2
/ 2/x pi\\ / 2/x pi\\
| cos |- - --|| | cos |- - --||
| \2 2 /| | \2 2 /| 2/x pi\
|1 - ------------| 4*|1 - ------------| *cos |- - --|
| 2/x\ | 6/x pi\ 4/x pi\ | 2/x\ | \2 2 /
| cos |-| | 64*cos |- - --| 16*cos |- - --| | cos |-| |
\ \2/ / \2 2 / \2 2 / \ \2/ /
------------------- - --------------------------- + --------------------------- - ----------------------------------
4 6 4 4
/ 2/x pi\\ / 2/x pi\\ / 2/x pi\\ / 2/x pi\\
| cos |- - --|| | cos |- - --|| | cos |- - --|| | cos |- - --||
| \2 2 /| | \2 2 /| 6/x\ | \2 2 /| 4/x\ | \2 2 /| 2/x\
|1 + ------------| |1 + ------------| *cos |-| |1 + ------------| *cos |-| |1 + ------------| *cos |-|
| 2/x\ | | 2/x\ | \2/ | 2/x\ | \2/ | 2/x\ | \2/
| cos |-| | | cos |-| | | cos |-| | | cos |-| |
\ \2/ / \ \2/ / \ \2/ / \ \2/ /
$$\frac{\left(1 - \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{4}}{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{4}} - \frac{4 \left(1 - \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{4} \cos^{2}{\left(\frac{x}{2} \right)}} + \frac{16 \cos^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{4} \cos^{4}{\left(\frac{x}{2} \right)}} - \frac{64 \cos^{6}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{6} \cos^{6}{\left(\frac{x}{2} \right)}}$$
4 2
/ 2/x\ \ / 2/x\ \
| sec |-| | | sec |-| |
| \2/ | | \2/ | 2/x\
|1 - ------------| 4*|1 - ------------| *sec |-|
| 2/x pi\| 6/x\ 4/x\ | 2/x pi\| \2/
| sec |- - --|| 64*sec |-| 16*sec |-| | sec |- - --||
\ \2 2 // \2/ \2/ \ \2 2 //
------------------- - -------------------------------- + -------------------------------- - --------------------------------
4 6 4 4
/ 2/x\ \ / 2/x\ \ / 2/x\ \ / 2/x\ \
| sec |-| | | sec |-| | | sec |-| | | sec |-| |
| \2/ | | \2/ | 6/x pi\ | \2/ | 4/x pi\ | \2/ | 2/x pi\
|1 + ------------| |1 + ------------| *sec |- - --| |1 + ------------| *sec |- - --| |1 + ------------| *sec |- - --|
| 2/x pi\| | 2/x pi\| \2 2 / | 2/x pi\| \2 2 / | 2/x pi\| \2 2 /
| sec |- - --|| | sec |- - --|| | sec |- - --|| | sec |- - --||
\ \2 2 // \ \2 2 // \ \2 2 // \ \2 2 //
$$\frac{\left(- \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{4}}{\left(\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{4}} - \frac{4 \left(- \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec^{2}{\left(\frac{x}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{4} \sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + \frac{16 \sec^{4}{\left(\frac{x}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{4} \sec^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - \frac{64 \sec^{6}{\left(\frac{x}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{6} \sec^{6}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}$$
4 2
/ 2/pi x\\ / 2/pi x\\
| csc |-- - -|| | csc |-- - -||
| \2 2/| | \2 2/| 2/pi x\
|1 - ------------| 4*|1 - ------------| *csc |-- - -|
| 2/x\ | 6/pi x\ 4/pi x\ | 2/x\ | \2 2/
| csc |-| | 64*csc |-- - -| 16*csc |-- - -| | csc |-| |
\ \2/ / \2 2/ \2 2/ \ \2/ /
------------------- - --------------------------- + --------------------------- - ----------------------------------
4 6 4 4
/ 2/pi x\\ / 2/pi x\\ / 2/pi x\\ / 2/pi x\\
| csc |-- - -|| | csc |-- - -|| | csc |-- - -|| | csc |-- - -||
| \2 2/| | \2 2/| 6/x\ | \2 2/| 4/x\ | \2 2/| 2/x\
|1 + ------------| |1 + ------------| *csc |-| |1 + ------------| *csc |-| |1 + ------------| *csc |-|
| 2/x\ | | 2/x\ | \2/ | 2/x\ | \2/ | 2/x\ | \2/
| csc |-| | | csc |-| | | csc |-| | | csc |-| |
\ \2/ / \ \2/ / \ \2/ / \ \2/ /
$$\frac{\left(1 - \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{4}}{\left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{4}} - \frac{4 \left(1 - \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{4} \csc^{2}{\left(\frac{x}{2} \right)}} + \frac{16 \csc^{4}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{4} \csc^{4}{\left(\frac{x}{2} \right)}} - \frac{64 \csc^{6}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{6} \csc^{6}{\left(\frac{x}{2} \right)}}$$
// 0 for x mod pi = 0\ // 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\ || | || | || | || | || | || 6/x\ | || 2/x\ | || 2 | || 4/x\ | || 4 | || 64*cot |-| | || 4*cot |-| | ||/ 2/x\\ | || 16*cot |-| | ||/ 2/x\\ | || \2/ | || \2/ | |||-1 + cot |-|| | || \2/ | |||-1 + cot |-|| | - |<-------------- otherwise | - |<-------------- otherwise |*|<\ \2// | + |<-------------- otherwise | + |<\ \2// | || 6 | || 2 | ||--------------- otherwise | || 4 | ||--------------- otherwise | ||/ 2/x\\ | ||/ 2/x\\ | || 2 | ||/ 2/x\\ | || 4 | |||1 + cot |-|| | |||1 + cot |-|| | || / 2/x\\ | |||1 + cot |-|| | || / 2/x\\ | ||\ \2// | ||\ \2// | || |1 + cot |-|| | ||\ \2// | || |1 + cot |-|| | \\ / \\ / \\ \ \2// / \\ / \\ \ \2// /
$$\left(- \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) - \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{64 \cot^{6}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{16 \cot^{4}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{4}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\ // 1 for x mod 2*pi = 0\
|| | || |
// 0 for x mod pi = 0\ // 0 for x mod pi = 0\ || 2 | // 0 for x mod pi = 0\ || 4 |
|| | || | ||/ 1 \ | || | ||/ 1 \ |
|| 64 | || 4 | |||-1 + -------| | || 16 | |||-1 + -------| |
||---------------------- otherwise | ||---------------------- otherwise | ||| 2/x\| | ||---------------------- otherwise | ||| 2/x\| |
|| 6 | || 2 | ||| tan |-|| | || 4 | ||| tan |-|| |
- |
$$\left(- \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \tan^{2}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)\right) - \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{64}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{6} \tan^{6}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{16}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{4} \tan^{4}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{4}}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{4}} & \text{otherwise} \end{cases}\right)$$
// / 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\ || 0 for |x + --| mod pi = 0|
|| | || | || \ 2 / | || | || \ 2 / |
|| 6/x\ | || 2/x\ | || | || 4/x\ | || |
|| 64*cot |-| | || 4*cot |-| | || 2/x pi\ | || 16*cot |-| | || 4/x pi\ |
|| \2/ | || \2/ | || 4*cot |- + --| | || \2/ | || 16*cot |- + --| |
- |<-------------- otherwise | - |<-------------- otherwise |*|< \2 4 / | + |<-------------- otherwise | + |< \2 4 / |
|| 6 | || 2 | ||------------------- otherwise | || 4 | ||------------------- otherwise |
||/ 2/x\\ | ||/ 2/x\\ | || 2 | ||/ 2/x\\ | || 4 |
|||1 + cot |-|| | |||1 + cot |-|| | ||/ 2/x pi\\ | |||1 + cot |-|| | ||/ 2/x pi\\ |
||\ \2// | ||\ \2// | |||1 + cot |- + --|| | ||\ \2// | |||1 + cot |- + --|| |
\\ / \\ / ||\ \2 4 // | \\ / ||\ \2 4 // |
\\ / \\ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) - \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{64 \cot^{6}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{16 \cot^{4}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{16 \cot^{4}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{4}} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\
// 0 for x mod pi = 0\ // 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ // 0 for x mod pi = 0\ || |
|| | || | || | || | || 4 |
|| 6/x\ | || 2/x\ | || 2 | || 8/x\ | || / 1 \ 8/x\ |
|| 64*tan |-| | || 4*tan |-| | ||/ 2/x\\ | || 4096*tan |-| | ||256*|-1 + -------| *tan |-| |
|| \2/ | || \2/ | |||1 - tan |-|| | || \4/ | || | 2/x\| \4/ |
- |<-------------- otherwise | - |<-------------- otherwise |*|<\ \2// | + |<---------------------- otherwise | + |< | tan |-|| |
|| 6 | || 2 | ||-------------- otherwise | || 8 | || \ \2// |
||/ 2/x\\ | ||/ 2/x\\ | || 2 | ||/ 2/x\\ 4/x\ | ||--------------------------- otherwise |
|||1 + tan |-|| | |||1 + tan |-|| | ||/ 2/x\\ | |||1 + tan |-|| *tan |-| | || 8 |
||\ \2// | ||\ \2// | |||1 + tan |-|| | ||\ \4// \2/ | || / 2/x\\ |
\\ / \\ / \\\ \2// / \\ / || |1 + tan |-|| |
\\ \ \4// /
$$\left(- \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) - \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{64 \tan^{6}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4096 \tan^{8}{\left(\frac{x}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{8} \tan^{4}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{256 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{4} \tan^{8}{\left(\frac{x}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{8}} & \text{otherwise} \end{cases}\right)$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\
// 0 for x mod pi = 0\ // 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ || | || |
|| | || | || | || // x \ | || // x \ |
||/ 0 for x mod pi = 0 | ||/ 0 for x mod pi = 0 | ||/ 1 for x mod 2*pi = 0 | || || 0 for - mod pi = 0| | || 4 || 0 for - mod pi = 0| |
- |<| | - |<| |*|<| | + |< 4/x\ || 2 | | + |
$$\left(- \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\16 \left(\begin{cases} 0 & \text{for}\: \frac{x}{2} \bmod \pi = 0 \\256 \sin^{16}{\left(\frac{x}{4} \right)} \cot^{8}{\left(\frac{x}{4} \right)} & \text{otherwise} \end{cases}\right) \cot^{4}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{6}{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{4} \left(\begin{cases} 0 & \text{for}\: \frac{x}{2} \bmod \pi = 0 \\256 \sin^{16}{\left(\frac{x}{4} \right)} \cot^{8}{\left(\frac{x}{4} \right)} & \text{otherwise} \end{cases}\right) & \text{otherwise} \end{cases}\right)$$
// / 3*pi\ \ // / 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\ || 1 for |x + ----| mod 2*pi = 0| || \ 2 / | || | || \ 2 / | || | || \ 2 / | || | || 2 | || | || 4 | || | || 6 | ||/ 2/x\\ | || 2 | ||/ 2/x\\ | || 4 | ||/ 2/x pi\\ | |||-1 + cot |-|| | ||/ 2/x pi\\ | |||-1 + cot |-|| | ||/ 2/x pi\\ | - |<|-1 + tan |- + --|| | - |<\ \2// |*|<|-1 + tan |- + --|| | + |<\ \2// | + |<|-1 + tan |- + --|| | ||\ \2 4 // | ||--------------- otherwise | ||\ \2 4 // | ||--------------- otherwise | ||\ \2 4 // | ||-------------------- otherwise | || 2 | ||-------------------- otherwise | || 4 | ||-------------------- otherwise | || 6 | || / 2/x\\ | || 2 | || / 2/x\\ | || 4 | ||/ 2/x pi\\ | || |1 + cot |-|| | ||/ 2/x pi\\ | || |1 + cot |-|| | ||/ 2/x pi\\ | |||1 + tan |- + --|| | \\ \ \2// / |||1 + tan |- + --|| | \\ \ \2// / |||1 + tan |- + --|| | \\\ \2 4 // / \\\ \2 4 // / \\\ \2 4 // /
$$\left(- \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{4}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)^{4}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{4}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)^{6}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases}\right)$$
// 0 for x mod pi = 0\ // 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\
|| 6 | || 2 | || | || 4 | || |
|| sin (x) | || sin (x) | || 2 | || sin (x) | || 4 |
||------------------------- otherwise | ||------------------------ otherwise | ||/ 2 4/x\\ | ||------------------------ otherwise | ||/ 2 4/x\\ |
|| 6 | || 2 | |||sin (x) - 4*sin |-|| | || 4 | |||sin (x) - 4*sin |-|| |
- |
$$\left(- \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{\sin^{2}{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(- 4 \sin^{4}{\left(\frac{x}{2} \right)} + \sin^{2}{\left(x \right)}\right)^{2}}{\left(4 \sin^{4}{\left(\frac{x}{2} \right)} + \sin^{2}{\left(x \right)}\right)^{2}} & \text{otherwise} \end{cases}\right)\right) - \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{\sin^{6}{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{6} \sin^{12}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{\sin^{4}{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{4} \sin^{8}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(- 4 \sin^{4}{\left(\frac{x}{2} \right)} + \sin^{2}{\left(x \right)}\right)^{4}}{\left(4 \sin^{4}{\left(\frac{x}{2} \right)} + \sin^{2}{\left(x \right)}\right)^{4}} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\ // 1 for x mod 2*pi = 0\
|| | || |
// 0 for x mod pi = 0\ // 0 for x mod pi = 0\ || 2 | // 0 for x mod pi = 0\ || 4 |
|| | || | ||/ 2 \ | || | ||/ 2 \ |
|| 6 | || 2 | ||| sin (x) | | || 4 | ||| sin (x) | |
|| sin (x) | || sin (x) | |||-1 + ---------| | || sin (x) | |||-1 + ---------| |
||------------------------- otherwise | ||------------------------ otherwise | ||| 4/x\| | ||------------------------ otherwise | ||| 4/x\| |
|| 6 | || 2 | ||| 4*sin |-|| | || 4 | ||| 4*sin |-|| |
- |
$$\left(- \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{\sin^{2}{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)\right) - \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{\sin^{6}{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{6} \sin^{12}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{\sin^{4}{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{4} \sin^{8}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{4}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{4}} & \text{otherwise} \end{cases}\right)$$
// x \ // x \ // x \
// x \ || 1 for - mod 2*pi = 0| || 1 for - mod 2*pi = 0| // x \ // x \ // x \ || 1 for - mod 2*pi = 0|
|| 0 for - mod pi = 0| || 2 | || 2 | || 0 for - mod pi = 0| || 0 for - mod pi = 0| || 0 for - mod pi = 0| || 2 |
|| 2 | || | || | || 2 | || 2 | || 2 | || |
- 68*|< | - 8*|< 6 | + 9*|< 8 | + 30*|< | + 39*|< | - 64*|< |*|< 6 |
|| 6/x\ 12/x\ | ||/ 2/x\\ 12/x\ | ||/ 2/x\\ 16/x\ | || 4/x\ 8/x\ | || 8/x\ 16/x\ | || 6/x\ 12/x\ | ||/ 2/x\\ 12/x\ |
||64*cot |-|*sin |-| otherwise | |||-1 + cot |-|| *sin |-| otherwise | |||-1 + cot |-|| *sin |-| otherwise | ||16*cot |-|*sin |-| otherwise | ||256*cot |-|*sin |-| otherwise | ||64*cot |-|*sin |-| otherwise | |||-1 + cot |-|| *sin |-| otherwise |
\\ \4/ \4/ / ||\ \4// \4/ | ||\ \4// \4/ | \\ \4/ \4/ / \\ \4/ \4/ / \\ \4/ \4/ / ||\ \4// \4/ |
\\ / \\ / \\ /
$$\left(- 64 \left(\begin{cases} 0 & \text{for}\: \frac{x}{2} \bmod \pi = 0 \\64 \sin^{12}{\left(\frac{x}{4} \right)} \cot^{6}{\left(\frac{x}{4} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \frac{x}{2} \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{6} \sin^{12}{\left(\frac{x}{4} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(30 \left(\begin{cases} 0 & \text{for}\: \frac{x}{2} \bmod \pi = 0 \\16 \sin^{8}{\left(\frac{x}{4} \right)} \cot^{4}{\left(\frac{x}{4} \right)} & \text{otherwise} \end{cases}\right)\right) - \left(68 \left(\begin{cases} 0 & \text{for}\: \frac{x}{2} \bmod \pi = 0 \\64 \sin^{12}{\left(\frac{x}{4} \right)} \cot^{6}{\left(\frac{x}{4} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(39 \left(\begin{cases} 0 & \text{for}\: \frac{x}{2} \bmod \pi = 0 \\256 \sin^{16}{\left(\frac{x}{4} \right)} \cot^{8}{\left(\frac{x}{4} \right)} & \text{otherwise} \end{cases}\right)\right) - \left(8 \left(\begin{cases} 1 & \text{for}\: \frac{x}{2} \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{6} \sin^{12}{\left(\frac{x}{4} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(9 \left(\begin{cases} 1 & \text{for}\: \frac{x}{2} \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{8} \sin^{16}{\left(\frac{x}{4} \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\
// 0 for x mod pi = 0\ // 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ || | || |
|| | || | || | || // x \ | || // x \ |
||/ 0 for x mod pi = 0 | ||/ 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 | |
||| 6/x\ | ||| 2/x\ | ||| 2 | || || | | || || | |
||| 64*cot |-| | ||| 4*cot |-| | |||/ 2/x\\ | || || 8/x\ | | || 4 || 8/x\ | |
- |<| \2/ | - |<| \2/ |*|<||-1 + cot |-|| | + |< 4/x\ || 256*cot |-| | | + |
$$\left(- \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\16 \left(\begin{cases} 0 & \text{for}\: \frac{x}{2} \bmod \pi = 0 \\\frac{256 \cot^{8}{\left(\frac{x}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{8}} & \text{otherwise} \end{cases}\right) \cot^{4}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{64 \cot^{6}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{4} \left(\begin{cases} 0 & \text{for}\: \frac{x}{2} \bmod \pi = 0 \\\frac{256 \cot^{8}{\left(\frac{x}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{8}} & \text{otherwise} \end{cases}\right) & \text{otherwise} \end{cases}\right)$$
// x \ // x \ // x \ // x \ // x \ // x \ // x \
|| 0 for - mod pi = 0| || 1 for - mod 2*pi = 0| || 1 for - mod 2*pi = 0| || 0 for - mod pi = 0| || 0 for - mod pi = 0| || 0 for - mod pi = 0| || 1 for - mod 2*pi = 0|
|| 2 | || 2 | || 2 | || 2 | || 2 | || 2 | || 2 |
|| | || | || | || | || | || | || |
|| 6/x\ | || 6 | || 8 | || 4/x\ | || 8/x\ | || 6/x\ | || 6 |
|| 64*cot |-| | ||/ 2/x\\ | ||/ 2/x\\ | || 16*cot |-| | || 256*cot |-| | || 64*cot |-| | ||/ 2/x\\ |
- 68*|< \4/ | - 8*|<|-1 + cot |-|| | + 9*|<|-1 + cot |-|| | + 30*|< \4/ | + 39*|< \4/ | - 64*|< \4/ |*|<|-1 + cot |-|| |
||-------------- otherwise | ||\ \4// | ||\ \4// | ||-------------- otherwise | ||-------------- otherwise | ||-------------- otherwise | ||\ \4// |
|| 6 | ||--------------- otherwise | ||--------------- otherwise | || 4 | || 8 | || 6 | ||--------------- otherwise |
||/ 2/x\\ | || 6 | || 8 | ||/ 2/x\\ | ||/ 2/x\\ | ||/ 2/x\\ | || 6 |
|||1 + cot |-|| | || / 2/x\\ | || / 2/x\\ | |||1 + cot |-|| | |||1 + cot |-|| | |||1 + cot |-|| | || / 2/x\\ |
||\ \4// | || |1 + cot |-|| | || |1 + cot |-|| | ||\ \4// | ||\ \4// | ||\ \4// | || |1 + cot |-|| |
\\ / \\ \ \4// / \\ \ \4// / \\ / \\ / \\ / \\ \ \4// /
$$\left(- 64 \left(\begin{cases} 0 & \text{for}\: \frac{x}{2} \bmod \pi = 0 \\\frac{64 \cot^{6}{\left(\frac{x}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \frac{x}{2} \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{6}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases}\right)\right) + \left(39 \left(\begin{cases} 0 & \text{for}\: \frac{x}{2} \bmod \pi = 0 \\\frac{256 \cot^{8}{\left(\frac{x}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{8}} & \text{otherwise} \end{cases}\right)\right) - \left(68 \left(\begin{cases} 0 & \text{for}\: \frac{x}{2} \bmod \pi = 0 \\\frac{64 \cot^{6}{\left(\frac{x}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases}\right)\right) + \left(30 \left(\begin{cases} 0 & \text{for}\: \frac{x}{2} \bmod \pi = 0 \\\frac{16 \cot^{4}{\left(\frac{x}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4}} & \text{otherwise} \end{cases}\right)\right) - \left(8 \left(\begin{cases} 1 & \text{for}\: \frac{x}{2} \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{6}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases}\right)\right) + \left(9 \left(\begin{cases} 1 & \text{for}\: \frac{x}{2} \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{8}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{8}} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for x mod 2*pi = 0\ // 1 for x mod 2*pi = 0\
|| | || |
// 0 for x mod pi = 0\ // 0 for x mod pi = 0\ || 2 | // 0 for x mod pi = 0\ || 4 |
|| | || | ||/ 2/x\ \ | || | ||/ 2/x\ \ |
|| 6/x\ | || 2/x\ | ||| cos |-| | | || 4/x\ | ||| cos |-| | |
|| 64*cos |-| | || 4*cos |-| | ||| \2/ | | || 16*cos |-| | ||| \2/ | |
|| \2/ | || \2/ | |||-1 + ------------| | || \2/ | |||-1 + ------------| |
||-------------------------------- otherwise | ||-------------------------------- otherwise | ||| 2/x pi\| | ||-------------------------------- otherwise | ||| 2/x pi\| |
|| 6 | || 2 | ||| cos |- - --|| | || 4 | ||| cos |- - --|| |
- |
$$\left(- \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \cos^{2}{\left(\frac{x}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) - \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{64 \cos^{6}{\left(\frac{x}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{6} \cos^{6}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{16 \cos^{4}{\left(\frac{x}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{4} \cos^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1\right)^{4}}{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{4}} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\ // 1 for x mod 2*pi = 0\
|| | || |
// 0 for x mod pi = 0\ // 0 for x mod pi = 0\ || 2 | // 0 for x mod pi = 0\ || 4 |
|| | || | ||/ 2/x pi\\ | || | ||/ 2/x pi\\ |
|| 6/x pi\ | || 2/x pi\ | ||| sec |- - --|| | || 4/x pi\ | ||| sec |- - --|| |
|| 64*sec |- - --| | || 4*sec |- - --| | ||| \2 2 /| | || 16*sec |- - --| | ||| \2 2 /| |
|| \2 2 / | || \2 2 / | |||-1 + ------------| | || \2 2 / | |||-1 + ------------| |
||--------------------------- otherwise | ||--------------------------- otherwise | ||| 2/x\ | | ||--------------------------- otherwise | ||| 2/x\ | |
|| 6 | || 2 | ||| sec |-| | | || 4 | ||| sec |-| | |
- |
$$\left(- \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \sec^{2}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)\right) - \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{64 \sec^{6}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{6} \sec^{6}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{16 \sec^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{4} \sec^{4}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{4}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{4}} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\ // 1 for x mod 2*pi = 0\
|| | || |
// 0 for x mod pi = 0\ // 0 for x mod pi = 0\ || 2 | // 0 for x mod pi = 0\ || 4 |
|| | || | ||/ 2/x\ \ | || | ||/ 2/x\ \ |
|| 6/x\ | || 2/x\ | ||| csc |-| | | || 4/x\ | ||| csc |-| | |
|| 64*csc |-| | || 4*csc |-| | ||| \2/ | | || 16*csc |-| | ||| \2/ | |
|| \2/ | || \2/ | |||-1 + ------------| | || \2/ | |||-1 + ------------| |
||-------------------------------- otherwise | ||-------------------------------- otherwise | ||| 2/pi x\| | ||-------------------------------- otherwise | ||| 2/pi x\| |
|| 6 | || 2 | ||| csc |-- - -|| | || 4 | ||| csc |-- - -|| |
- |
$$\left(- \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \csc^{2}{\left(\frac{x}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2} \csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) - \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{64 \csc^{6}{\left(\frac{x}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1\right)^{6} \csc^{6}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{16 \csc^{4}{\left(\frac{x}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1\right)^{4} \csc^{4}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - 1\right)^{4}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1\right)^{4}} & \text{otherwise} \end{cases}\right)$$
-Piecewise((0, Mod(x = pi, 0)), (64*csc(x/2)^6/((1 + csc(x/2)^2/csc(pi/2 - x/2)^2)^6*csc(pi/2 - x/2)^6), True)) - 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))
Численный ответ
[src]
cos(x)^4 + sin(x)^4 - sin(x)^6 - cos(x)^2*sin(x)^2
cos(x)^4 + sin(x)^4 - sin(x)^6 - cos(x)^2*sin(x)^2