Господин Экзамен
Другие калькуляторы
-
Как пользоваться?
- 2*sin(a)-2*cos(a) если a=3
- 4*sin(x)^2+4*cos(x)^2 если x=-1/2
- 48*m^2-n^2 если m=1/4
- 49-14*y+y^2 если y=1/3
- Разложить многочлен на множители:
- s^2-k^2+18*s+81
- x^8-х^3
- x^2-x-12
- n^2+n+1
- Общий знаменатель:
- cos(pi/4-a)-cos(pi/4+a)
- ((c)/(c^2+3*c+9))-((1)/(c-3))+((27)/(c^3-27))
- (4^(n+2)-4^n)/15^(n+1)*5^n/12^-n*2^(-4*n+3)
- (x+5)/(x+7)+14/(x+14)
- Умножение столбиком:
- 18*27
- 25*24
- 15*10
- 12*50
- Деление столбиком:
- 213/4
- 800/3
- 3900/2
- 78/5
- Раскрыть скобки в:
- 15*(2+b)+3*(12-b)
- a^3*(-a^2)
- (t+r)^8*(t+r)^14
- 9*(a-2*b)-10*(3*a+b)
- Разложение числа на простые множители:
- 1584
- 10800
- 11111
- 2544
- График функции y =:
- -1/2
- Интеграл d{x}:
-
-1/2
- Производная:
- -1/2
-
Идентичные выражения
- четыре *sin(x)^ два + четыре *cos(x)^ два если x=- один / два
- 4 умножить на синус от (x) в квадрате плюс 4 умножить на косинус от (x) в квадрате если x равно минус 1 делить на 2
- четыре умножить на синус от (x) в степени два плюс четыре умножить на косинус от (x) в степени два если x равно минус один делить на два
- 4*sin(x)2+4*cos(x)2 если x=-1/2
- 4*sinx2+4*cosx2 если x=-1/2
- 4*sin(x)²+4*cos(x)² если x=-1/2
- 4*sin(x) в степени 2+4*cos(x) в степени 2 если x=-1/2
- 4sin(x)^2+4cos(x)^2 если x=-1/2
- 4sin(x)2+4cos(x)2 если x=-1/2
- 4sinx2+4cosx2 если x=-1/2
- 4sinx^2+4cosx^2 если x=-1/2
- 4*sin(x)^2+4*cos(x)^2 при x=-1/2
- 4*sin(x)^2+4*cos(x)^2 если x=-1 разделить на 2
-
Похожие выражения
- 4*sin(x)^2-4*cos(x)^2 если x=-1/2
- 4*sin(x)^2+4*cos(x)^2 если x=+1/2
- 4*sinx^2+4*cosx^2 если x=-1/2
4*sin(x)^2+4*cos(x)^2 если x=-1/2
Решение
Вы ввели
[src]
2 2 4*sin (x) + 4*cos (x)
$$4 \sin^{2}{\left(x \right)} + 4 \cos^{2}{\left(x \right)}$$
4*sin(x)^2 + 4*cos(x)^2
Объединение рациональных выражений
[src]
/ 2 2 \ 4*\cos (x) + sin (x)/
$$4 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)$$
4*(cos(x)^2 + sin(x)^2)
Степени
[src]
2
2 / I*x -I*x\
/ -I*x I*x\ |e e |
- \- e + e / + 4*|---- + -----|
\ 2 2 /
$$4 \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{2} - \left(e^{i x} - e^{- i x}\right)^{2}$$
-(-exp(-i*x) + exp(i*x))^2 + 4*(exp(i*x)/2 + exp(-i*x)/2)^2
Тригонометрическая часть
[src]
4
$$4$$
2 2/ pi\
4*cos (x) + 4*cos |x - --|
\ 2 /
$$4 \cos^{2}{\left(x \right)} + 4 \cos^{2}{\left(x - \frac{\pi}{2} \right)}$$
4 4 ------- + ------- 2 2 csc (x) sec (x)
$$\frac{4}{\sec^{2}{\left(x \right)}} + \frac{4}{\csc^{2}{\left(x \right)}}$$
2 2/ pi\
4*sin (x) + 4*sin |x + --|
\ 2 /
$$4 \sin^{2}{\left(x \right)} + 4 \sin^{2}{\left(x + \frac{\pi}{2} \right)}$$
4 4
------- + ------------
2 2/ pi\
sec (x) sec |x - --|
\ 2 /
$$\frac{4}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} + \frac{4}{\sec^{2}{\left(x \right)}}$$
4 4
------- + ------------
2 2/pi \
csc (x) csc |-- - x|
\2 /
$$\frac{4}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} + \frac{4}{\csc^{2}{\left(x \right)}}$$
4 4
------- + ------------
2 2/pi \
sec (x) sec |-- - x|
\2 /
$$\frac{4}{\sec^{2}{\left(- x + \frac{\pi}{2} \right)}} + \frac{4}{\sec^{2}{\left(x \right)}}$$
4 4
------------ + ------------
2 2/pi \
csc (pi - x) csc |-- - x|
\2 /
$$\frac{4}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} + \frac{4}{\csc^{2}{\left(- x + \pi \right)}}$$
2 10 - 8*cos(x) - 4*(1 - cos(x)) + 2*cos(2*x)
$$- 4 \left(- \cos{\left(x \right)} + 1\right)^{2} - 8 \cos{\left(x \right)} + 2 \cos{\left(2 x \right)} + 10$$
2
/ 2/x pi\\ 2
2*(1 + cos(2*x)) + |1 - cot |- + --|| *(1 + sin(x))
\ \2 4 //
$$\left(- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\sin{\left(x \right)} + 1\right)^{2} + 2 \left(\cos{\left(2 x \right)} + 1\right)$$
2/x pi\
16*tan |- + --|
\2 4 /
2*(1 - cos(2*x)) + -------------------
2
/ 2/x pi\\
|1 + tan |- + --||
\ \2 4 //
$$2 \cdot \left(- \cos{\left(2 x \right)} + 1\right) + \frac{16 \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
2
/ 2/x\\ 2/x\
4*|1 - tan |-|| 16*tan |-|
\ \2// \2/
---------------- + --------------
2 2
/ 2/x\\ / 2/x\\
|1 + tan |-|| |1 + tan |-||
\ \2// \ \2//
$$\frac{4 \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + \frac{16 \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$
2/x\ 2/x pi\
16*cot |-| 16*tan |- + --|
\2/ \2 4 /
-------------- + -------------------
2 2
/ 2/x\\ / 2/x pi\\
|1 + cot |-|| |1 + tan |- + --||
\ \2// \ \2 4 //
$$\frac{16 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + \frac{16 \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
2/x\ 2/x pi\
16*tan |-| 16*tan |- + --|
\2/ \2 4 /
-------------- + -------------------
2 2
/ 2/x\\ / 2/x pi\\
|1 + tan |-|| |1 + tan |- + --||
\ \2// \ \2 4 //
$$\frac{16 \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} + \frac{16 \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$
2
/ 1 \
4*|1 - -------|
| 2/x\|
| cot |-||
\ \2// 16
---------------- + ----------------------
2 2
/ 1 \ / 1 \ 2/x\
|1 + -------| |1 + -------| *cot |-|
| 2/x\| | 2/x\| \2/
| cot |-|| | cot |-||
\ \2// \ \2//
$$\frac{4 \left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{2}} + \frac{16}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \cot^{2}{\left(\frac{x}{2} \right)}}$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ || | || | 4*|< 2 | + 4*|< 2 | ||sin (x) otherwise | ||cos (x) otherwise | \\ / \\ /
$$\left(4 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(4 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right)$$
2 2
/ 2/x\\ / 2/x pi\\
4*|-1 + cot |-|| 4*|-1 + tan |- + --||
\ \2// \ \2 4 //
----------------- + ----------------------
2 2
/ 2/x\\ / 2/x pi\\
|1 + cot |-|| |1 + tan |- + --||
\ \2// \ \2 4 //
$$\frac{4 \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} + \frac{4 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$
2 2
/ 2/x pi\\ / 2/x\\
4*|1 - cot |- + --|| 4*|1 - tan |-||
\ \2 4 // \ \2//
--------------------- + ----------------
2 2
/ 2/x pi\\ / 2/x\\
|1 + cot |- + --|| |1 + tan |-||
\ \2 4 // \ \2//
$$\frac{4 \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + \frac{4 \left(- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ || | || | 4*|< 2/ pi\ | + 4*|< 2 | ||cos |x - --| otherwise | ||cos (x) otherwise | \\ \ 2 / / \\ /
$$\left(4 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\cos^{2}{\left(x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(4 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ || | || | 4*|< 2 | + 4*|< 2/ pi\ | ||sin (x) otherwise | ||sin |x + --| otherwise | \\ / \\ \ 2 / /
$$\left(4 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(4 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\sin^{2}{\left(x + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ || | || | || 1 | || 1 | 4*|<------------ otherwise | + 4*|<------- otherwise | || 2/ pi\ | || 2 | ||sec |x - --| | ||sec (x) | \\ \ 2 / / \\ /
$$\left(4 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(4 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{1}{\sec^{2}{\left(x \right)}} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ || | || | || 1 | || 1 | 4*|<------- otherwise | + 4*|<------------ otherwise | || 2 | || 2/pi \ | ||csc (x) | ||csc |-- - x| | \\ / \\ \2 / /
$$\left(4 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\csc^{2}{\left(x \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(4 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{1}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right)$$
// / 3*pi\ \
// 1 for x mod 2*pi = 0\ || 1 for |x + ----| mod 2*pi = 0|
|| | || \ 2 / |
4*|< 2 | + 4*|< |
||cos (x) otherwise | || 4/x\ 2/x\ |
\\ / ||- 4*cos |-| + 4*cos |-| otherwise |
\\ \2/ \2/ /
$$\left(4 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(4 \left(\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\- 4 \cos^{4}{\left(\frac{x}{2} \right)} + 4 \cos^{2}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right)\right)$$
2
/ 4/x\\
| 4*sin |-||
| \2/|
4*|1 - ---------| 4/x\
| 2 | 64*sin |-|
\ sin (x) / \2/
------------------ + ------------------------
2 2
/ 4/x\\ / 4/x\\
| 4*sin |-|| | 4*sin |-||
| \2/| | \2/| 2
|1 + ---------| |1 + ---------| *sin (x)
| 2 | | 2 |
\ sin (x) / \ sin (x) /
$$\frac{4 \left(- \frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{2}}{\left(\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{2}} + \frac{64 \sin^{4}{\left(\frac{x}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{2} \sin^{2}{\left(x \right)}}$$
// / pi\ \
// 0 for x mod pi = 0\ || 0 for |x + --| mod pi = 0|
|| | || \ 2 / |
4*|< 2 | + 4*|< |
||sin (x) otherwise | || 2 2/x pi\ |
\\ / ||(1 + sin(x)) *cot |- + --| otherwise |
\\ \2 4 / /
$$\left(4 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(4 \left(\begin{cases} 0 & \text{for}\: \left(x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(x \right)} + 1\right)^{2} \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 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 | 4*|<| | + 4*|<| | ||< 2 otherwise | ||< 2 otherwise | |||sin (x) otherwise | |||cos (x) otherwise | \\\ / \\\ /
$$\left(4 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(4 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ || | || | || 2/x\ | || 2 | || 4*cot |-| | ||/ 2/x\\ | || \2/ | |||-1 + cot |-|| | 4*|<-------------- otherwise | + 4*|<\ \2// | || 2 | ||--------------- otherwise | ||/ 2/x\\ | || 2 | |||1 + cot |-|| | || / 2/x\\ | ||\ \2// | || |1 + cot |-|| | \\ / \\ \ \2// /
$$\left(4 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(4 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ || | || | || 2/x\ | || 2 | || 4*tan |-| | ||/ 2/x\\ | || \2/ | |||1 - tan |-|| | 4*|<-------------- otherwise | + 4*|<\ \2// | || 2 | ||-------------- otherwise | ||/ 2/x\\ | || 2 | |||1 + tan |-|| | ||/ 2/x\\ | ||\ \2// | |||1 + tan |-|| | \\ / \\\ \2// /
$$\left(4 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(4 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right)$$
2
/ 2/x pi\\
| cos |- - --||
| \2 2 /|
4*|1 - ------------|
| 2/x\ | 2/x pi\
| cos |-| | 16*cos |- - --|
\ \2/ / \2 2 /
--------------------- + ---------------------------
2 2
/ 2/x pi\\ / 2/x pi\\
| cos |- - --|| | cos |- - --||
| \2 2 /| | \2 2 /| 2/x\
|1 + ------------| |1 + ------------| *cos |-|
| 2/x\ | | 2/x\ | \2/
| cos |-| | | cos |-| |
\ \2/ / \ \2/ /
$$\frac{4 \left(1 - \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{2}}{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{2}} + \frac{16 \cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \cos^{2}{\left(\frac{x}{2} \right)}}$$
2
/ 2/x\ \
| sec |-| |
| \2/ |
4*|1 - ------------|
| 2/x pi\| 2/x\
| sec |- - --|| 16*sec |-|
\ \2 2 // \2/
--------------------- + --------------------------------
2 2
/ 2/x\ \ / 2/x\ \
| sec |-| | | sec |-| |
| \2/ | | \2/ | 2/x pi\
|1 + ------------| |1 + ------------| *sec |- - --|
| 2/x pi\| | 2/x pi\| \2 2 /
| sec |- - --|| | sec |- - --||
\ \2 2 // \ \2 2 //
$$\frac{4 \left(- \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}}{\left(\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}} + \frac{16 \sec^{2}{\left(\frac{x}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}$$
2
/ 2/pi x\\
| csc |-- - -||
| \2 2/|
4*|1 - ------------|
| 2/x\ | 2/pi x\
| csc |-| | 16*csc |-- - -|
\ \2/ / \2 2/
--------------------- + ---------------------------
2 2
/ 2/pi x\\ / 2/pi x\\
| csc |-- - -|| | csc |-- - -||
| \2 2/| | \2 2/| 2/x\
|1 + ------------| |1 + ------------| *csc |-|
| 2/x\ | | 2/x\ | \2/
| csc |-| | | csc |-| |
\ \2/ / \ \2/ /
$$\frac{4 \left(1 - \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{2}}{\left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{2}} + \frac{16 \csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \csc^{2}{\left(\frac{x}{2} \right)}}$$
// 1 for x mod 2*pi = 0\
|| |
// 0 for x mod pi = 0\ || 2 |
|| | ||/ 1 \ |
|| 4 | |||-1 + -------| |
||---------------------- otherwise | ||| 2/x\| |
|| 2 | ||| tan |-|| |
4*|
$$\left(4 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \tan^{2}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(4 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)\right)$$
// / pi\ \
// 0 for x mod pi = 0\ || 0 for |x + --| mod pi = 0|
|| | || \ 2 / |
|| 2/x\ | || |
|| 4*cot |-| | || 2/x pi\ |
|| \2/ | || 4*cot |- + --| |
4*|<-------------- otherwise | + 4*|< \2 4 / |
|| 2 | ||------------------- otherwise |
||/ 2/x\\ | || 2 |
|||1 + cot |-|| | ||/ 2/x pi\\ |
||\ \2// | |||1 + cot |- + --|| |
\\ / ||\ \2 4 // |
\\ /
$$\left(4 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(4 \left(\begin{cases} 0 & \text{for}\: \left(x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right)$$
// / 3*pi\ \
// 1 for x mod 2*pi = 0\ || 1 for |x + ----| mod 2*pi = 0|
|| | || \ 2 / |
|| 2 | || |
||/ 2/x\\ | || 2 |
|||-1 + cot |-|| | ||/ 2/x pi\\ |
4*|<\ \2// | + 4*|<|-1 + tan |- + --|| |
||--------------- otherwise | ||\ \2 4 // |
|| 2 | ||-------------------- otherwise |
|| / 2/x\\ | || 2 |
|| |1 + cot |-|| | ||/ 2/x pi\\ |
\\ \ \2// / |||1 + tan |- + --|| |
\\\ \2 4 // /
$$\left(4 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(4 \left(\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for x mod pi = 0\
|| | // 1 for x mod 2*pi = 0\
|| 2 | || |
|| sin (x) | || 2 |
||------------------------ otherwise | ||/ 2 4/x\\ |
|| 2 | |||sin (x) - 4*sin |-|| |
4*|
$$\left(4 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{\sin^{2}{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(4 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(- 4 \sin^{4}{\left(\frac{x}{2} \right)} + \sin^{2}{\left(x \right)}\right)^{2}}{\left(4 \sin^{4}{\left(\frac{x}{2} \right)} + \sin^{2}{\left(x \right)}\right)^{2}} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for x mod 2*pi = 0\
|| |
// 0 for x mod pi = 0\ || 2 |
|| | ||/ 2 \ |
|| 2 | ||| sin (x) | |
|| sin (x) | |||-1 + ---------| |
||------------------------ otherwise | ||| 4/x\| |
|| 2 | ||| 4*sin |-|| |
4*|
$$\left(4 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{\sin^{2}{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(4 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ || | || | ||/ 0 for x mod pi = 0 | ||/ 1 for x mod 2*pi = 0 | ||| | ||| | ||| 2/x\ | ||| 2 | ||| 4*cot |-| | |||/ 2/x\\ | 4*|<| \2/ | + 4*|<||-1 + cot |-|| | ||<-------------- otherwise otherwise | ||<\ \2// otherwise | ||| 2 | |||--------------- otherwise | |||/ 2/x\\ | ||| 2 | ||||1 + cot |-|| | ||| / 2/x\\ | |||\ \2// | ||| |1 + cot |-|| | \\\ / \\\ \ \2// /
$$\left(4 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(4 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for x mod 2*pi = 0\
|| |
// 0 for x mod pi = 0\ || 2 |
|| | ||/ 2/x\ \ |
|| 2/x\ | ||| cos |-| | |
|| 4*cos |-| | ||| \2/ | |
|| \2/ | |||-1 + ------------| |
||-------------------------------- otherwise | ||| 2/x pi\| |
|| 2 | ||| cos |- - --|| |
4*|
$$\left(4 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \cos^{2}{\left(\frac{x}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(4 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for x mod 2*pi = 0\
|| |
// 0 for x mod pi = 0\ || 2 |
|| | ||/ 2/x pi\\ |
|| 2/x pi\ | ||| sec |- - --|| |
|| 4*sec |- - --| | ||| \2 2 /| |
|| \2 2 / | |||-1 + ------------| |
||--------------------------- otherwise | ||| 2/x\ | |
|| 2 | ||| sec |-| | |
4*|
$$\left(4 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \sec^{2}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(4 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for x mod 2*pi = 0\
|| |
// 0 for x mod pi = 0\ || 2 |
|| | ||/ 2/x\ \ |
|| 2/x\ | ||| csc |-| | |
|| 4*csc |-| | ||| \2/ | |
|| \2/ | |||-1 + ------------| |
||-------------------------------- otherwise | ||| 2/pi x\| |
|| 2 | ||| csc |-- - -|| |
4*|
$$\left(4 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \csc^{2}{\left(\frac{x}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2} \csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(4 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right)$$
4*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)) + 4*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))