Господин Экзамен
Другие калькуляторы
-
Как пользоваться?
- sin(2*x)*cos(2*x) если x=3/2
- 10*x^3+x^2+10*x+1 если x=-3/2
- 1/(s+1)/(s^2+s) если s=-1/2
- 8*x^3+y^3+6*y^2+12*y+8 если y=2
- Разложить многочлен на множители:
- x^2-10*x+16
- 9*a^2*b^2-12*a*b^3
- 6*c^2*d^2+54*c^2*d^3+9*c*d^9
- 64-9*a^2
- Общий знаменатель:
- a^2+5*a/x^2+a*x^6/3*a+15
- (m+n-4*m*n/(m+n))/(m/(m+n)-n/(n-n)-2*m*n/(m^2-n^2))
- n^2-3*n/64*n^2-1/n4-27*n/64*n^2+16*n+1
- m-3*n/m+n*m^2-n^2/3*m-9*n
- Умножение столбиком:
- 13*25
- 15*15
- 17*30
- 40*12
- Деление столбиком:
- 54/8
- 44/3
- 88/4
- 87/9
- Раскрыть скобки в:
- 8*(6*x-7)-17*x
- 16*u^2-8*u*(3*v+1)+(3*v+1)^2
- 5*(4-2*c)
- (x-2)^2+(x-1)*(x+1)
- Разложение числа на простые множители:
- 52668
- 65
- 6552
- 28800
- График функции y =:
- 3/2
- Интеграл d{x}:
-
3/2
- Производная:
- 3/2
-
Идентичные выражения
- sin(два *x)*cos(два *x) если x= три / два
- синус от (2 умножить на x) умножить на косинус от (2 умножить на x) если x равно 3 делить на 2
- синус от (два умножить на x) умножить на косинус от (два умножить на x) если x равно три делить на два
- sin(2x)cos(2x) если x=3/2
- sin2xcos2x если x=3/2
- sin(2*x)*cos(2*x) при x=3/2
- sin(2*x)*cos(2*x) если x=3 разделить на 2
sin(2*x)*cos(2*x) если x=3/2
Решение
Вы ввели
[src]
sin(2*x)*cos(2*x)
$$\sin{\left(2 x \right)} \cos{\left(2 x \right)}$$
sin(2*x)*cos(2*x)
Подстановка условия
[src]
sin(2*x)*cos(2*x) при x = 3/2
подставляем
sin(2*x)*cos(2*x)
$$\sin{\left(2 x \right)} \cos{\left(2 x \right)}$$
sin(4*x) -------- 2
$$\frac{\sin{\left(4 x \right)}}{2}$$
переменные
x = 3/2
$$x = \frac{3}{2}$$
sin(4*(3/2))
------------
2
$$\frac{\sin{\left(4 (3/2) \right)}}{2}$$
sin(6) ------ 2
$$\frac{\sin{\left(6 \right)}}{2}$$
sin(6)/2
Степени
[src]
/ -2*I*x 2*I*x\
|e e | / -2*I*x 2*I*x\
-I*|------- + ------|*\- e + e /
\ 2 2 /
-------------------------------------------
2
$$- \frac{i \left(\frac{e^{2 i x}}{2} + \frac{e^{- 2 i x}}{2}\right) \left(e^{2 i x} - e^{- 2 i x}\right)}{2}$$
-i*(exp(-2*i*x)/2 + exp(2*i*x)/2)*(-exp(-2*i*x) + exp(2*i*x))/2
Тригонометрическая часть
[src]
sin(4*x) -------- 2
$$\frac{\sin{\left(4 x \right)}}{2}$$
1 ---------- 2*csc(4*x)
$$\frac{1}{2 \csc{\left(4 x \right)}}$$
/ pi\
cos|4*x - --|
\ 2 /
-------------
2
$$\frac{\cos{\left(4 x - \frac{\pi}{2} \right)}}{2}$$
1
---------------
/ pi\
2*sec|4*x - --|
\ 2 /
$$\frac{1}{2 \sec{\left(4 x - \frac{\pi}{2} \right)}}$$
1 ----------------- csc(2*x)*sec(2*x)
$$\frac{1}{\csc{\left(2 x \right)} \sec{\left(2 x \right)}}$$
/ pi\
cos(2*x)*cos|2*x - --|
\ 2 /
$$\cos{\left(2 x \right)} \cos{\left(2 x - \frac{\pi}{2} \right)}$$
/pi \
sin(2*x)*sin|-- + 2*x|
\2 /
$$\sin{\left(2 x \right)} \sin{\left(2 x + \frac{\pi}{2} \right)}$$
tan(2*x)
-------------
2
1 + tan (2*x)
$$\frac{\tan{\left(2 x \right)}}{\tan^{2}{\left(2 x \right)} + 1}$$
/ 2 \ \-1 + 2*cos (x)/*sin(2*x)
$$\left(2 \cos^{2}{\left(x \right)} - 1\right) \sin{\left(2 x \right)}$$
1
----------------------
/ pi\
sec(2*x)*sec|2*x - --|
\ 2 /
$$\frac{1}{\sec{\left(2 x \right)} \sec{\left(2 x - \frac{\pi}{2} \right)}}$$
1
----------------------
/pi \
csc(2*x)*csc|-- - 2*x|
\2 /
$$\frac{1}{\csc{\left(2 x \right)} \csc{\left(- 2 x + \frac{\pi}{2} \right)}}$$
1
----------------------
/pi \
sec(2*x)*sec|-- - 2*x|
\2 /
$$\frac{1}{\sec{\left(2 x \right)} \sec{\left(- 2 x + \frac{\pi}{2} \right)}}$$
1
---------------------------
/pi \
csc(pi - 2*x)*csc|-- - 2*x|
\2 /
$$\frac{1}{\csc{\left(- 2 x + \pi \right)} \csc{\left(- 2 x + \frac{\pi}{2} \right)}}$$
/ 2 \ 2*\-1 + 2*cos (x)/*cos(x)*sin(x)
$$2 \cdot \left(2 \cos^{2}{\left(x \right)} - 1\right) \sin{\left(x \right)} \cos{\left(x \right)}$$
/ 2 2 \ 2*\cos (x) - sin (x)/*cos(x)*sin(x)
$$2 \left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)}$$
/ 0 for 4*x mod pi = 0
<
\sin(4*x) otherwise
-----------------------------
2
$$\frac{\begin{cases} 0 & \text{for}\: 4 x \bmod \pi = 0 \\\sin{\left(4 x \right)} & \text{otherwise} \end{cases}}{2}$$
/ 2 \
2*\1 - tan (x)/*tan(x)
----------------------
2
/ 2 \
\1 + tan (x)/
$$\frac{2 \cdot \left(- \tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{\left(\tan^{2}{\left(x \right)} + 1\right)^{2}}$$
/ 1 \
2*|1 - -------|
| 2 |
\ cot (x)/
---------------------
2
/ 1 \
|1 + -------| *cot(x)
| 2 |
\ cot (x)/
$$\frac{2 \cdot \left(1 - \frac{1}{\cot^{2}{\left(x \right)}}\right)}{\left(1 + \frac{1}{\cot^{2}{\left(x \right)}}\right)^{2} \cot{\left(x \right)}}$$
/ 0 for 4*x mod pi = 0
|
| 2*cot(2*x)
<------------- otherwise
| 2
|1 + cot (2*x)
\
----------------------------------
2
$$\frac{\begin{cases} 0 & \text{for}\: 4 x \bmod \pi = 0 \\\frac{2 \cot{\left(2 x \right)}}{\cot^{2}{\left(2 x \right)} + 1} & \text{otherwise} \end{cases}}{2}$$
/ pi\
4*tan(x)*tan|x + --|
\ 4 /
--------------------------------
/ 2 \ / 2/ pi\\
\1 + tan (x)/*|1 + tan |x + --||
\ \ 4 //
$$\frac{4 \tan{\left(x \right)} \tan{\left(x + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(x \right)} + 1\right) \left(\tan^{2}{\left(x + \frac{\pi}{4} \right)} + 1\right)}$$
/ pi\
4*cot(x)*tan|x + --|
\ 4 /
--------------------------------
/ 2 \ / 2/ pi\\
\1 + cot (x)/*|1 + tan |x + --||
\ \ 4 //
$$\frac{4 \tan{\left(x + \frac{\pi}{4} \right)} \cot{\left(x \right)}}{\left(\tan^{2}{\left(x + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right)}$$
/ 2 \ / 2/ pi\\
\-1 + cot (x)/*|-1 + tan |x + --||
\ \ 4 //
----------------------------------
/ 2 \ / 2/ pi\\
\1 + cot (x)/*|1 + tan |x + --||
\ \ 4 //
$$\frac{\left(\tan^{2}{\left(x + \frac{\pi}{4} \right)} - 1\right) \left(\cot^{2}{\left(x \right)} - 1\right)}{\left(\tan^{2}{\left(x + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right)}$$
/ 2/ pi\\ / 2 \ |1 - cot |x + --||*\1 - tan (x)/ \ \ 4 // -------------------------------- / 2/ pi\\ / 2 \ |1 + cot |x + --||*\1 + tan (x)/ \ \ 4 //
$$\frac{\left(- \tan^{2}{\left(x \right)} + 1\right) \left(- \cot^{2}{\left(x + \frac{\pi}{4} \right)} + 1\right)}{\left(\tan^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(x + \frac{\pi}{4} \right)} + 1\right)}$$
/ 4 \
2 | 4*sin (x)|
4*sin (x)*|1 - ---------|
| 2 |
\ sin (2*x)/
-------------------------
2
/ 4 \
| 4*sin (x)|
|1 + ---------| *sin(2*x)
| 2 |
\ sin (2*x)/
$$\frac{4 \left(- \frac{4 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} + 1\right) \sin^{2}{\left(x \right)}}{\left(\frac{4 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} + 1\right)^{2} \sin{\left(2 x \right)}}$$
// 0 for 2*x mod pi = 0\ // 1 for x mod pi = 0\ |< |*|< | \\sin(2*x) otherwise / \\cos(2*x) otherwise /
$$\left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\cos{\left(2 x \right)} & \text{otherwise} \end{cases}\right)$$
// 0 for 2*x mod pi = 0\ || | // 1 for x mod pi = 0\ |< / pi\ |*|< | ||cos|2*x - --| otherwise | \\cos(2*x) otherwise / \\ \ 2 / /
$$\left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\cos{\left(2 x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\cos{\left(2 x \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod pi = 0\
// 0 for 2*x mod pi = 0\ || |
|< |*|< /pi \ |
\\sin(2*x) otherwise / ||sin|-- + 2*x| otherwise |
\\ \2 / /
$$\left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(2 x + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)$$
/ 2 \
| sec (x) |
2*|1 - ------------|*sec(x)
| 2/ pi\|
| sec |x - --||
\ \ 2 //
-------------------------------
2
/ 2 \
| sec (x) | / pi\
|1 + ------------| *sec|x - --|
| 2/ pi\| \ 2 /
| sec |x - --||
\ \ 2 //
$$\frac{2 \left(- \frac{\sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(x \right)}}{\left(\frac{\sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec{\left(x - \frac{\pi}{2} \right)}}$$
/ 2/ pi\\
| cos |x - --||
| \ 2 /| / pi\
2*|1 - ------------|*cos|x - --|
| 2 | \ 2 /
\ cos (x) /
--------------------------------
2
/ 2/ pi\\
| cos |x - --||
| \ 2 /|
|1 + ------------| *cos(x)
| 2 |
\ cos (x) /
$$\frac{2 \cdot \left(1 - \frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}}\right) \cos{\left(x - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}}\right)^{2} \cos{\left(x \right)}}$$
// /pi \ \
// 0 for 2*x mod pi = 0\ || 0 for |-- + 2*x| mod pi = 0|
|< |*|< \2 / |
\\sin(2*x) otherwise / || |
\\cos(2*x) otherwise /
$$\left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(2 x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\cos{\left(2 x \right)} & \text{otherwise} \end{cases}\right)$$
// / 3*pi\ \
// 1 for x mod pi = 0\ || 1 for |2*x + ----| mod 2*pi = 0|
|< |*|< \ 2 / |
\\cos(2*x) otherwise / || |
\\sin(2*x) otherwise /
$$\left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\cos{\left(2 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(2 x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases}\right)$$
// 0 for 2*x mod pi = 0\ || | // 1 for x mod pi = 0\ || 1 | || | |<------------- otherwise |*|< 1 | || / pi\ | ||-------- otherwise | ||sec|2*x - --| | \\sec(2*x) / \\ \ 2 / /
$$\left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{1}{\sec{\left(2 x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\sec{\left(2 x \right)}} & \text{otherwise} \end{cases}\right)$$
/ 2/pi \\
| csc |-- - x||
| \2 /| /pi \
2*|1 - ------------|*csc|-- - x|
| 2 | \2 /
\ csc (x) /
--------------------------------
2
/ 2/pi \\
| csc |-- - x||
| \2 /|
|1 + ------------| *csc(x)
| 2 |
\ csc (x) /
$$\frac{2 \cdot \left(1 - \frac{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}}\right) \csc{\left(- x + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}}\right)^{2} \csc{\left(x \right)}}$$
// 1 for x mod pi = 0\
// 0 for 2*x mod pi = 0\ || |
|| | || 1 |
|< 1 |*|<------------- otherwise |
||-------- otherwise | || /pi \ |
\\csc(2*x) / ||csc|-- - 2*x| |
\\ \2 / /
$$\left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{1}{\csc{\left(2 x \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// 0 for 2*x mod pi = 0\ // 1 for x mod pi = 0\ || | || | || 2*cot(x) | || 2 | |<----------- otherwise |*|<-1 + cot (x) | || 2 | ||------------ otherwise | ||1 + cot (x) | || 2 | \\ / \\1 + cot (x) /
$$\left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// 0 for 2*x mod pi = 0\ // 1 for x mod pi = 0\ || | || | || 2*tan(x) | || 2 | |<----------- otherwise |*|<1 - tan (x) | || 2 | ||----------- otherwise | ||1 + tan (x) | || 2 | \\ / \\1 + tan (x) /
$$\left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2 \tan{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{- \tan^{2}{\left(x \right)} + 1}{\tan^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod pi = 0\
// 0 for 2*x mod pi = 0\ || |
|| | || 1 |
|| 2 | ||-1 + ------- |
||-------------------- otherwise | || 2 |
|
$$\left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(x \right)}}\right) \tan{\left(x \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(x \right)}}}{1 + \frac{1}{\tan^{2}{\left(x \right)}}} & \text{otherwise} \end{cases}\right)$$
// /pi \ \
|| 0 for |-- + 2*x| mod pi = 0|
// 0 for 2*x mod pi = 0\ || \2 / |
|| | || |
|| 2*cot(x) | || / pi\ |
|<----------- otherwise |*|< 2*cot|x + --| |
|| 2 | || \ 4 / |
||1 + cot (x) | ||---------------- otherwise |
\\ / || 2/ pi\ |
||1 + cot |x + --| |
\\ \ 4 / /
$$\left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(2 x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(x + \frac{\pi}{4} \right)}}{\cot^{2}{\left(x + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// / 3*pi\ \
|| 1 for |2*x + ----| mod 2*pi = 0|
// 1 for x mod pi = 0\ || \ 2 / |
|| | || |
|| 2 | || 2/ pi\ |
|<-1 + cot (x) |*|<-1 + tan |x + --| |
||------------ otherwise | || \ 4 / |
|| 2 | ||----------------- otherwise |
\\1 + cot (x) / || 2/ pi\ |
|| 1 + tan |x + --| |
\\ \ 4 / /
$$\left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(2 x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(x + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(x + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// 0 for 2*x mod pi = 0\ // 1 for x mod pi = 0\
|| | || |
|
$$\left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\cos{\left(2 x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod pi = 0\
|| |
// 0 for 2*x mod pi = 0\ || 2 |
|| | || sin (2*x) |
|| sin(2*x) | ||-1 + --------- |
||----------------------- otherwise | || 4 |
|
$$\left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{\sin{\left(2 x \right)}}{\left(1 + \frac{\sin^{2}{\left(2 x \right)}}{4 \sin^{4}{\left(x \right)}}\right) \sin^{2}{\left(x \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(2 x \right)}}{4 \sin^{4}{\left(x \right)}}}{1 + \frac{\sin^{2}{\left(2 x \right)}}{4 \sin^{4}{\left(x \right)}}} & \text{otherwise} \end{cases}\right)$$
// 0 for 2*x mod pi = 0\ // 1 for x mod pi = 0\ || | || | ||/ 0 for 2*x mod pi = 0 | ||/ 1 for x mod pi = 0 | ||| | ||| | |<| 2*cot(x) |*|<| 2 | ||<----------- otherwise otherwise | ||<-1 + cot (x) otherwise | ||| 2 | |||------------ otherwise | |||1 + cot (x) | ||| 2 | \\\ / \\\1 + cot (x) /
$$\left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod pi = 0\
|| |
// 0 for 2*x mod pi = 0\ || 2 |
|| | || cos (x) |
|| 2*cos(x) | ||-1 + ------------ |
||------------------------------ otherwise | || 2/ pi\ |
||/ 2 \ | || cos |x - --| |
|<| cos (x) | / pi\ |*|< \ 2 / |
|||1 + ------------|*cos|x - --| | ||----------------- otherwise |
||| 2/ pi\| \ 2 / | || 2 |
||| cos |x - --|| | || cos (x) |
||\ \ 2 // | || 1 + ------------ |
\\ / || 2/ pi\ |
|| cos |x - --| |
\\ \ 2 / /
$$\left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2 \cos{\left(x \right)}}{\left(\frac{\cos^{2}{\left(x \right)}}{\cos^{2}{\left(x - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{\frac{\cos^{2}{\left(x \right)}}{\cos^{2}{\left(x - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(x \right)}}{\cos^{2}{\left(x - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod pi = 0\
// 0 for 2*x mod pi = 0\ || |
|| | || 2/ pi\ |
|| / pi\ | || sec |x - --| |
|| 2*sec|x - --| | || \ 2 / |
|| \ 2 / | ||-1 + ------------ |
||------------------------- otherwise | || 2 |
|
$$\left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2 \sec{\left(x - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(x - \frac{\pi}{2} \right)}}{\sec^{2}{\left(x \right)}}\right) \sec{\left(x \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(x - \frac{\pi}{2} \right)}}{\sec^{2}{\left(x \right)}}}{1 + \frac{\sec^{2}{\left(x - \frac{\pi}{2} \right)}}{\sec^{2}{\left(x \right)}}} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod pi = 0\
|| |
// 0 for 2*x mod pi = 0\ || 2 |
|| | || csc (x) |
|| 2*csc(x) | ||-1 + ------------ |
||------------------------------ otherwise | || 2/pi \ |
||/ 2 \ | || csc |-- - x| |
|<| csc (x) | /pi \ |*|< \2 / |
|||1 + ------------|*csc|-- - x| | ||----------------- otherwise |
||| 2/pi \| \2 / | || 2 |
||| csc |-- - x|| | || csc (x) |
||\ \2 // | || 1 + ------------ |
\\ / || 2/pi \ |
|| csc |-- - x| |
\\ \2 / /
$$\left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2 \csc{\left(x \right)}}{\left(\frac{\csc^{2}{\left(x \right)}}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{\frac{\csc^{2}{\left(x \right)}}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(x \right)}}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)$$
Piecewise((0, Mod(2*x = pi, 0)), (2*csc(x)/((1 + csc(x)^2/csc(pi/2 - x)^2)*csc(pi/2 - x)), True))*Piecewise((1, Mod(x = pi, 0)), ((-1 + csc(x)^2/csc(pi/2 - x)^2)/(1 + csc(x)^2/csc(pi/2 - x)^2), True))