Господин Экзамен
cot(x)*tan(x)-(-cos(x))*(-sin(x)) если x=3
Решение
Вы ввели
[src]
cot(x)*tan(x) - -cos(x)*-sin(x)
$$- - \sin{\left(x \right)} \left(- \cos{\left(x \right)}\right) + \tan{\left(x \right)} \cot{\left(x \right)}$$
cot(x)*tan(x) - (-cos(x))*(-sin(x))
Общее упрощение
[src]
sin(2*x)
1 - --------
2
$$- \frac{\sin{\left(2 x \right)}}{2} + 1$$
1 - sin(2*x)/2
Подстановка условия
[src]
cot(x)*tan(x) - (-cos(x))*(-sin(x)) при x = 3
подставляем
cot(x)*tan(x) - -cos(x)*-sin(x)
$$- - \sin{\left(x \right)} \left(- \cos{\left(x \right)}\right) + \tan{\left(x \right)} \cot{\left(x \right)}$$
sin(2*x)
1 - --------
2
$$- \frac{\sin{\left(2 x \right)}}{2} + 1$$
переменные
x = 3
$$x = 3$$
sin(2*(3))
1 - ----------
2
$$- \frac{\sin{\left(2 (3) \right)}}{2} + 1$$
sin(2*3)
1 - --------
2
$$- \frac{\sin{\left(2 \cdot 3 \right)}}{2} + 1$$
sin(6)
1 - ------
2
$$- \frac{\sin{\left(6 \right)}}{2} + 1$$
1 - sin(6)/2
Объединение рациональных выражений
[src]
cot(x)*tan(x) - cos(x)*sin(x)
$$- \sin{\left(x \right)} \cos{\left(x \right)} + \tan{\left(x \right)} \cot{\left(x \right)}$$
cot(x)*tan(x) - cos(x)*sin(x)
Рациональный знаменатель
[src]
cot(x)*tan(x) - cos(x)*sin(x)
$$- \sin{\left(x \right)} \cos{\left(x \right)} + \tan{\left(x \right)} \cot{\left(x \right)}$$
cot(x)*tan(x) - cos(x)*sin(x)
Степени
[src]
cot(x)*tan(x) - cos(x)*sin(x)
$$- \sin{\left(x \right)} \cos{\left(x \right)} + \tan{\left(x \right)} \cot{\left(x \right)}$$
/ I*x -I*x\
/ -I*x I*x\ | e e |
I*\- e + e /*|- ---- - -----| / I*x -I*x\
\ 2 2 / I*\- e + e /*cot(x)
- ----------------------------------- + -------------------------
2 I*x -I*x
e + e
$$- \frac{i \left(- \frac{e^{i x}}{2} - \frac{e^{- i x}}{2}\right) \left(e^{i x} - e^{- i x}\right)}{2} + \frac{i \left(- e^{i x} + e^{- i x}\right) \cot{\left(x \right)}}{e^{i x} + e^{- i x}}$$
-i*(-exp(-i*x) + exp(i*x))*(-exp(i*x)/2 - exp(-i*x)/2)/2 + i*(-exp(i*x) + exp(-i*x))*cot(x)/(exp(i*x) + exp(-i*x))
Общий знаменатель
[src]
cot(x)*tan(x) - cos(x)*sin(x)
$$- \sin{\left(x \right)} \cos{\left(x \right)} + \tan{\left(x \right)} \cot{\left(x \right)}$$
cot(x)*tan(x) - cos(x)*sin(x)
Комбинаторика
[src]
cot(x)*tan(x) - cos(x)*sin(x)
$$- \sin{\left(x \right)} \cos{\left(x \right)} + \tan{\left(x \right)} \cot{\left(x \right)}$$
cot(x)*tan(x) - cos(x)*sin(x)
Собрать выражение
[src]
sin(2*x)
1 - --------
2
$$- \frac{\sin{\left(2 x \right)}}{2} + 1$$
cot(x)*tan(x) - cos(x)*sin(x)
$$- \sin{\left(x \right)} \cos{\left(x \right)} + \tan{\left(x \right)} \cot{\left(x \right)}$$
cot(x)*tan(x) - cos(x)*sin(x)
Тригонометрическая часть
[src]
1
1 - ----------
2*csc(2*x)
$$1 - \frac{1}{2 \csc{\left(2 x \right)}}$$
sin(2*x)
1 - --------
2
$$- \frac{\sin{\left(2 x \right)}}{2} + 1$$
1 - cos(x)*sin(x)
$$- \sin{\left(x \right)} \cos{\left(x \right)} + 1$$
/ pi\
cos|2*x - --|
\ 2 /
1 - -------------
2
$$- \frac{\cos{\left(2 x - \frac{\pi}{2} \right)}}{2} + 1$$
1
1 - ---------------
/ pi\
2*sec|2*x - --|
\ 2 /
$$1 - \frac{1}{2 \sec{\left(2 x - \frac{\pi}{2} \right)}}$$
1
1 - -------------
csc(x)*sec(x)
$$1 - \frac{1}{\csc{\left(x \right)} \sec{\left(x \right)}}$$
tan(x)
1 - -----------
2
1 + tan (x)
$$1 - \frac{\tan{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1}$$
/ pi\
1 - cos(x)*cos|x - --|
\ 2 /
$$- \cos{\left(x \right)} \cos{\left(x - \frac{\pi}{2} \right)} + 1$$
/ pi\
1 - sin(x)*sin|x + --|
\ 2 /
$$- \sin{\left(x \right)} \sin{\left(x + \frac{\pi}{2} \right)} + 1$$
1
1 - ------------------
/ pi\
sec(x)*sec|x - --|
\ 2 /
$$1 - \frac{1}{\sec{\left(x \right)} \sec{\left(x - \frac{\pi}{2} \right)}}$$
1
1 - ------------------
/pi \
csc(x)*csc|-- - x|
\2 /
$$1 - \frac{1}{\csc{\left(x \right)} \csc{\left(- x + \frac{\pi}{2} \right)}}$$
1
1 - ------------------
/pi \
sec(x)*sec|-- - x|
\2 /
$$1 - \frac{1}{\sec{\left(x \right)} \sec{\left(- x + \frac{\pi}{2} \right)}}$$
1
1 - -----------------------
/pi \
csc(pi - x)*csc|-- - x|
\2 /
$$1 - \frac{1}{\csc{\left(- x + \pi \right)} \csc{\left(- x + \frac{\pi}{2} \right)}}$$
cot(x)*tan(x) - cos(x)*sin(x)
$$- \sin{\left(x \right)} \cos{\left(x \right)} + \tan{\left(x \right)} \cot{\left(x \right)}$$
/x\
1 - (1 + cos(x))*cos(x)*tan|-|
\2/
$$- \left(\cos{\left(x \right)} + 1\right) \cos{\left(x \right)} \tan{\left(\frac{x}{2} \right)} + 1$$
/ 0 for 2*x mod pi = 0
<
\sin(2*x) otherwise
1 - -----------------------------
2
$$\left(- \frac{\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases}}{2}\right) + 1$$
/ 2/x\\ /x\
2*|1 - tan |-||*tan|-|
\ \2// \2/
1 - ----------------------
2
/ 2/x\\
|1 + tan |-||
\ \2//
$$- \frac{2 \cdot \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \tan{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + 1$$
/ 0 for 2*x mod pi = 0
|
| 2*cot(x)
<----------- otherwise
| 2
|1 + cot (x)
\
1 - --------------------------------
2
$$\left(- \frac{\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}}{2}\right) + 1$$
/ 1 \
2*|1 - -------|
| 2/x\|
| cot |-||
\ \2//
1 - ---------------------
2
/ 1 \ /x\
|1 + -------| *cot|-|
| 2/x\| \2/
| cot |-||
\ \2//
$$1 - \frac{2 \cdot \left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \cot{\left(\frac{x}{2} \right)}}$$
/x\ /x pi\
4*cot|-|*tan|- + --|
\2/ \2 4 /
1 - --------------------------------
/ 2/x\\ / 2/x pi\\
|1 + cot |-||*|1 + tan |- + --||
\ \2// \ \2 4 //
$$1 - \frac{4 \tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)} \cot{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
/x\ /x pi\
4*tan|-|*tan|- + --|
\2/ \2 4 /
1 - --------------------------------
/ 2/x\\ / 2/x pi\\
|1 + tan |-||*|1 + tan |- + --||
\ \2// \ \2 4 //
$$1 - \frac{4 \tan{\left(\frac{x}{2} \right)} \tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)}$$
2/x\ / 2/x pi\\ / 2/x\\
cos |-|*|1 - cot |- + --||*|1 - tan |-||*(1 + sin(x))
\2/ \ \2 4 // \ \2//
1 - -----------------------------------------------------
2
$$- \frac{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(x \right)} + 1\right) \cos^{2}{\left(\frac{x}{2} \right)}}{2} + 1$$
2/x\ / 2 4/x\\
4*sin |-|*|sin (x) - 4*sin |-||*sin(x)
\2/ \ \2//
1 - --------------------------------------
2
/ 2 4/x\\
|sin (x) + 4*sin |-||
\ \2//
$$- \frac{4 \left(- 4 \sin^{4}{\left(\frac{x}{2} \right)} + \sin^{2}{\left(x \right)}\right) \sin^{2}{\left(\frac{x}{2} \right)} \sin{\left(x \right)}}{\left(4 \sin^{4}{\left(\frac{x}{2} \right)} + \sin^{2}{\left(x \right)}\right)^{2}} + 1$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\
1 - |< |*|< |
\\sin(x) otherwise / \\cos(x) otherwise /
$$\left(- \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + 1$$
/ 2/x\\ / 2/x pi\\
|-1 + cot |-||*|-1 + tan |- + --||
\ \2// \ \2 4 //
1 - ----------------------------------
/ 2/x\\ / 2/x pi\\
|1 + cot |-||*|1 + tan |- + --||
\ \2// \ \2 4 //
$$- \frac{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} + 1$$
/ 4/x\\
| 4*sin |-||
2/x\ | \2/|
4*sin |-|*|1 - ---------|
\2/ | 2 |
\ sin (x) /
1 - -------------------------
2
/ 4/x\\
| 4*sin |-||
| \2/|
|1 + ---------| *sin(x)
| 2 |
\ sin (x) /
$$- \frac{4 \left(- \frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right) \sin^{2}{\left(\frac{x}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{2} \sin{\left(x \right)}} + 1$$
/ 2/x pi\\ / 2/x\\
|1 - cot |- + --||*|1 - tan |-||
\ \2 4 // \ \2//
1 - --------------------------------
/ 2/x pi\\ / 2/x\\
|1 + cot |- + --||*|1 + tan |-||
\ \2 4 // \ \2//
$$- \frac{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)} + 1$$
// 1 for x mod 2*pi = 0\
// 0 for x mod pi = 0\ || |
1 - |< |*|< / pi\ |
\\sin(x) otherwise / ||sin|x + --| otherwise |
\\ \ 2 / /
$$\left(- \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\sin{\left(x + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + 1$$
// 0 for x mod pi = 0\
|| | // 1 for x mod 2*pi = 0\
1 - |< / pi\ |*|< |
||cos|x - --| otherwise | \\cos(x) otherwise /
\\ \ 2 / /
$$\left(- \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\cos{\left(x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + 1$$
// / 3*pi\ \
// 1 for x mod 2*pi = 0\ || 1 for |x + ----| mod 2*pi = 0|
1 - |< |*|< \ 2 / |
\\cos(x) otherwise / || |
\\sin(x) otherwise /
$$\left(- \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + 1$$
// 0 for x mod pi = 0\
|| | // 1 for x mod 2*pi = 0\
|| 1 | || |
1 - |<----------- otherwise |*|< 1 |
|| / pi\ | ||------ otherwise |
||sec|x - --| | \\sec(x) /
\\ \ 2 / /
$$\left(- \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(x \right)}} & \text{otherwise} \end{cases}\right)\right) + 1$$
// 1 for x mod 2*pi = 0\
// 0 for x mod pi = 0\ || |
|| | || 1 |
1 - |< 1 |*|<----------- otherwise |
||------ otherwise | || /pi \ |
\\csc(x) / ||csc|-- - x| |
\\ \2 / /
$$\left(- \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\csc{\left(x \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + 1$$
// 0 for x mod pi = 0\
|| |
||1 - cos(x) | // 1 for x mod 2*pi = 0\
1 - |<---------- otherwise |*|< |
|| /x\ | \\cos(x) otherwise /
|| tan|-| |
\\ \2/ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{- \cos{\left(x \right)} + 1}{\tan{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + 1$$
/ 2/x\ \
| sec |-| |
| \2/ | /x\
2*|1 - ------------|*sec|-|
| 2/x pi\| \2/
| sec |- - --||
\ \2 2 //
1 - -------------------------------
2
/ 2/x\ \
| sec |-| |
| \2/ | /x pi\
|1 + ------------| *sec|- - --|
| 2/x pi\| \2 2 /
| sec |- - --||
\ \2 2 //
$$1 - \frac{2 \left(- \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{x}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}$$
/ 2/x pi\\
| cos |- - --||
| \2 2 /| /x pi\
2*|1 - ------------|*cos|- - --|
| 2/x\ | \2 2 /
| cos |-| |
\ \2/ /
1 - --------------------------------
2
/ 2/x pi\\
| cos |- - --||
| \2 2 /| /x\
|1 + ------------| *cos|-|
| 2/x\ | \2/
| cos |-| |
\ \2/ /
$$1 - \frac{2 \cdot \left(1 - \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right) \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \cos{\left(\frac{x}{2} \right)}}$$
/ 2/pi x\\
| csc |-- - -||
| \2 2/| /pi x\
2*|1 - ------------|*csc|-- - -|
| 2/x\ | \2 2/
| csc |-| |
\ \2/ /
1 - --------------------------------
2
/ 2/pi x\\
| csc |-- - -||
| \2 2/| /x\
|1 + ------------| *csc|-|
| 2/x\ | \2/
| csc |-| |
\ \2/ /
$$1 - \frac{2 \cdot \left(1 - \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right) \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \csc{\left(\frac{x}{2} \right)}}$$
// / pi\ \
|| 0 for |x + --| mod pi = 0|
// 0 for x mod pi = 0\ || \ 2 / |
1 - |< |*|< |
\\sin(x) otherwise / || /x pi\ |
||(1 + sin(x))*cot|- + --| otherwise |
\\ \2 4 / /
$$\left(- \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(x \right)} + 1\right) \cot{\left(\frac{x}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)\right) + 1$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\
|| | || |
|| /x\ | || 2/x\ |
|| 2*cot|-| | ||-1 + cot |-| |
1 - |< \2/ |*|< \2/ |
||----------- otherwise | ||------------ otherwise |
|| 2/x\ | || 2/x\ |
||1 + cot |-| | ||1 + cot |-| |
\\ \2/ / \\ \2/ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + 1$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\
|| | || |
|| /x\ | || 2/x\ |
|| 2*tan|-| | ||1 - tan |-| |
1 - |< \2/ |*|< \2/ |
||----------- otherwise | ||----------- otherwise |
|| 2/x\ | || 2/x\ |
||1 + tan |-| | ||1 + tan |-| |
\\ \2/ / \\ \2/ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{x}{2} \right)} + 1}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + 1$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\
|| | || |
1 - |
$$\left(- \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + 1$$
// 1 for x mod 2*pi = 0\
|| |
// 0 for x mod pi = 0\ || 1 |
|| | ||-1 + ------- |
|| 2 | || 2/x\ |
||-------------------- otherwise | || tan |-| |
1 - |
$$\left(- \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right) \tan{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right) + 1$$
// / pi\ \
// 0 for x mod pi = 0\ || 0 for |x + --| mod pi = 0|
|| | || \ 2 / |
|| /x\ | || |
|| 2*cot|-| | || /x pi\ |
1 - |< \2/ |*|< 2*cot|- + --| |
||----------- otherwise | || \2 4 / |
|| 2/x\ | ||---------------- otherwise |
||1 + cot |-| | || 2/x pi\ |
\\ \2/ / ||1 + cot |- + --| |
\\ \2 4 / /
$$\left(- \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + 1$$
// / 3*pi\ \
// 1 for x mod 2*pi = 0\ || 1 for |x + ----| mod 2*pi = 0|
|| | || \ 2 / |
|| 2/x\ | || |
||-1 + cot |-| | || 2/x pi\ |
1 - |< \2/ |*|<-1 + tan |- + --| |
||------------ otherwise | || \2 4 / |
|| 2/x\ | ||----------------- otherwise |
||1 + cot |-| | || 2/x pi\ |
\\ \2/ / || 1 + tan |- + --| |
\\ \2 4 / /
$$\left(- \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + 1$$
// 0 for x mod pi = 0\
|| |
|| 2*sin(x) | // 1 for x mod 2*pi = 0\
||---------------------------- otherwise | || |
|| / 2 \ | || 2 |
1 - |< | sin (x) | |*|< -4 + 4*sin (x) + 4*cos(x) |
||(1 - cos(x))*|1 + ---------| | ||--------------------------- otherwise |
|| | 4/x\| | || 2 2 |
|| | 4*sin |-|| | \\2*(1 - cos(x)) + 2*sin (x) /
|| \ \2// |
\\ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \sin{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right) \left(- \cos{\left(x \right)} + 1\right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{4 \sin^{2}{\left(x \right)} + 4 \cos{\left(x \right)} - 4}{2 \left(- \cos{\left(x \right)} + 1\right)^{2} + 2 \sin^{2}{\left(x \right)}} & \text{otherwise} \end{cases}\right)\right) + 1$$
// 1 for x mod 2*pi = 0\
|| |
// 0 for x mod pi = 0\ || 2 |
|| | || sin (x) |
|| sin(x) | ||-1 + --------- |
||----------------------- otherwise | || 4/x\ |
||/ 2 \ | || 4*sin |-| |
1 - |<| sin (x) | 2/x\ |*|< \2/ |
|||1 + ---------|*sin |-| | ||-------------- otherwise |
||| 4/x\| \2/ | || 2 |
||| 4*sin |-|| | || sin (x) |
||\ \2// | ||1 + --------- |
\\ / || 4/x\ |
|| 4*sin |-| |
\\ \2/ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{\sin{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right) \sin^{2}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}}{1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right) + 1$$
// 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\ | ||| 2/x\ |
1 - |<| 2*cot|-| |*|<|-1 + cot |-| |
||< \2/ otherwise | ||< \2/ otherwise |
|||----------- otherwise | |||------------ otherwise |
||| 2/x\ | ||| 2/x\ |
|||1 + cot |-| | |||1 + cot |-| |
\\\ \2/ / \\\ \2/ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + 1$$
// 1 for x mod 2*pi = 0\
|| |
// 0 for x mod pi = 0\ || 2/x\ |
|| | || cos |-| |
|| /x\ | || \2/ |
|| 2*cos|-| | ||-1 + ------------ |
|| \2/ | || 2/x pi\ |
||------------------------------ otherwise | || cos |- - --| |
1 - |
$$\left(- \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{x}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right) + 1$$
// 1 for x mod 2*pi = 0\
|| |
// 0 for x mod pi = 0\ || 2/x pi\ |
|| | || sec |- - --| |
|| /x pi\ | || \2 2 / |
|| 2*sec|- - --| | ||-1 + ------------ |
|| \2 2 / | || 2/x\ |
||------------------------- otherwise | || sec |-| |
1 - |
$$\left(- \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right) \sec{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right) + 1$$
// 1 for x mod 2*pi = 0\
|| |
// 0 for x mod pi = 0\ || 2/x\ |
|| | || csc |-| |
|| /x\ | || \2/ |
|| 2*csc|-| | ||-1 + ------------ |
|| \2/ | || 2/pi x\ |
||------------------------------ otherwise | || csc |-- - -| |
1 - |
$$\left(- \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{x}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right) + 1$$
1 - Piecewise((0, Mod(x = pi, 0)), (2*csc(x/2)/((1 + csc(x/2)^2/csc(pi/2 - x/2)^2)*csc(pi/2 - x/2)), True))*Piecewise((1, Mod(x = 2*pi, 0)), ((-1 + csc(x/2)^2/csc(pi/2 - x/2)^2)/(1 + csc(x/2)^2/csc(pi/2 - x/2)^2), True))