Господин Экзамен
Другие калькуляторы
-
Как пользоваться?
- sin(x+y)*sin(x)*sin(y) если y=1/4
- 3*(12+m) если m=1/2
- 3*x+5*x+17*x если x=3/2
- 3*a^2+7*a-6 если a=2
- Разложить многочлен на множители:
- c^6*d^3-k^3
- 9*x^2-4*y+4*y-1
- x^2-14*x+33
- 225*a^4-30*n^2*a^2+n^4
- Общий знаменатель:
- 4/y-2/(y-5)+2*y/(25-y^2)-10/(y^2-25)
- t^2/(t+14)+28*t+196/(t+14)
- (15*x-12)/x-(15*t-7)/t
- sin(2)*(pi-t)+sin(2)*(pi/2-t)/sin(pi-t)
- Умножение столбиком:
- 97*4090
- 32*52
- 745*500
- 10*63
- Деление столбиком:
- 864/7
- 23067/9
- 93748/9
- 25618/76
- Раскрыть скобки в:
- 125*8*a-55*a*18
- (x-1)*(x^2+x+1)-x*(x^2+2)
- 16*(x-y)^2-25*(x+y)^2
- 8*a-1+6+(3*a-7)*(-2)
- Разложение числа на простые множители:
- 3033
- 2117
- 7488
- 63973
- Производная:
- 1/4
- Интеграл d{x}:
-
1/4
- График функции y =:
- 1/4
-
Идентичные выражения
- sin(x+y)*sin(x)*sin(y) если y= один / четыре
- синус от (x плюс y) умножить на синус от (x) умножить на синус от (y) если y равно 1 делить на 4
- синус от (x плюс y) умножить на синус от (x) умножить на синус от (y) если y равно один делить на четыре
- sin(x+y)sin(x)sin(y) если y=1/4
- sinx+ysinxsiny если y=1/4
- sin(x+y)*sin(x)*sin(y) при y=1/4
- sin(x+y)*sin(x)*sin(y) если y=1 разделить на 4
-
Похожие выражения
- sin(x-y)*sin(x)*sin(y) если y=1/4
- sin(x+y)*sinx*sin(y) если y=1/4
sin(x+y)*sin(x)*sin(y) если y=1/4
Решение
Вы ввели
[src]
sin(x + y)*sin(x)*sin(y)
$$\sin{\left(x \right)} \sin{\left(y \right)} \sin{\left(x + y \right)}$$
sin(x + y)*sin(x)*sin(y)
Подстановка условия
[src]
sin(x + y)*sin(x)*sin(y) при y = 1/4
подставляем
sin(x + y)*sin(x)*sin(y)
$$\sin{\left(x \right)} \sin{\left(y \right)} \sin{\left(x + y \right)}$$
sin(x)*sin(y)*sin(x + y)
$$\sin{\left(x \right)} \sin{\left(y \right)} \sin{\left(x + y \right)}$$
переменные
y = 1/4
$$y = \frac{1}{4}$$
sin(x)*sin((1/4))*sin(x + (1/4))
$$\sin{\left((1/4) \right)} \sin{\left(x \right)} \sin{\left((1/4) + x \right)}$$
sin(x)*sin(1/4)*sin(x + 1/4)
$$\sin{\left(\frac{1}{4} \right)} \sin{\left(x \right)} \sin{\left(x + \frac{1}{4} \right)}$$
sin(1/4)*sin(x)*sin(1/4 + x)
$$\sin{\left(\frac{1}{4} \right)} \sin{\left(x \right)} \sin{\left(x + \frac{1}{4} \right)}$$
sin(1/4)*sin(x)*sin(1/4 + x)
Собрать выражение
[src]
sin(2*x + 2*y) sin(2*x) sin(2*y)
- -------------- + -------- + --------
4 4 4
$$\frac{\sin{\left(2 x \right)}}{4} + \frac{\sin{\left(2 y \right)}}{4} - \frac{\sin{\left(2 x + 2 y \right)}}{4}$$
-sin(2*x + 2*y)/4 + sin(2*x)/4 + sin(2*y)/4
Тригонометрическая часть
[src]
1 ------------------------ csc(x)*csc(y)*csc(x + y)
$$\frac{1}{\csc{\left(x \right)} \csc{\left(y \right)} \csc{\left(x + y \right)}}$$
1 --------------------------------------- csc(pi - x)*csc(pi - y)*csc(pi - x - y)
$$\frac{1}{\csc{\left(- x + \pi \right)} \csc{\left(- y + \pi \right)} \csc{\left(- x - y + \pi \right)}}$$
/ pi\ / pi\ / pi\ cos|x - --|*cos|y - --|*cos|x + y - --| \ 2 / \ 2 / \ 2 /
$$\cos{\left(x - \frac{\pi}{2} \right)} \cos{\left(y - \frac{\pi}{2} \right)} \cos{\left(x + y - \frac{\pi}{2} \right)}$$
sin(2*(x + y)) sin(2*x) sin(2*y)
- -------------- + -------- + --------
4 4 4
$$\frac{\sin{\left(2 x \right)}}{4} + \frac{\sin{\left(2 y \right)}}{4} - \frac{\sin{\left(2 \left(x + y\right) \right)}}{4}$$
1 --------------------------------------- / pi\ / pi\ / pi\ sec|x - --|*sec|y - --|*sec|x + y - --| \ 2 / \ 2 / \ 2 /
$$\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)} \sec{\left(y - \frac{\pi}{2} \right)} \sec{\left(x + y - \frac{\pi}{2} \right)}}$$
1 --------------------------------------- /pi \ /pi \ /pi \ sec|-- - x|*sec|-- - y|*sec|-- - x - y| \2 / \2 / \2 /
$$\frac{1}{\sec{\left(- x + \frac{\pi}{2} \right)} \sec{\left(- y + \frac{\pi}{2} \right)} \sec{\left(- x - y + \frac{\pi}{2} \right)}}$$
2/x\ 2/y\ /x\ /y\ /x y\
8*cos |-|*cos |-|*tan|-|*tan|-|*tan|- + -|
\2/ \2/ \2/ \2/ \2 2/
------------------------------------------
2/x y\
1 + tan |- + -|
\2 2/
$$\frac{8 \cos^{2}{\left(\frac{x}{2} \right)} \cos^{2}{\left(\frac{y}{2} \right)} \tan{\left(\frac{x}{2} \right)} \tan{\left(\frac{y}{2} \right)} \tan{\left(\frac{x}{2} + \frac{y}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} + \frac{y}{2} \right)} + 1}$$
/x\ /y\ /x\ /y\ /x + y\
4*(1 + cos(x + y))*cos|-|*cos|-|*sin|-|*sin|-|*sin|-----|
\2/ \2/ \2/ \2/ \ 2 /
---------------------------------------------------------
/x + y\
cos|-----|
\ 2 /
$$\frac{4 \left(\cos{\left(x + y \right)} + 1\right) \sin{\left(\frac{x}{2} \right)} \sin{\left(\frac{y}{2} \right)} \sin{\left(\frac{x + y}{2} \right)} \cos{\left(\frac{x}{2} \right)} \cos{\left(\frac{y}{2} \right)}}{\cos{\left(\frac{x + y}{2} \right)}}$$
/x\ /y\ /x y\ /x\ /y\ /x + y\
4*(1 + cos(x + y))*cos|-|*cos|-|*sec|- + -|*sin|-|*sin|-|*sin|-----|
\2/ \2/ \2 2/ \2/ \2/ \ 2 /
$$4 \left(\cos{\left(x + y \right)} + 1\right) \sin{\left(\frac{x}{2} \right)} \sin{\left(\frac{y}{2} \right)} \sin{\left(\frac{x + y}{2} \right)} \cos{\left(\frac{x}{2} \right)} \cos{\left(\frac{y}{2} \right)} \sec{\left(\frac{x}{2} + \frac{y}{2} \right)}$$
/x\ /y\ /x y\
8*tan|-|*tan|-|*tan|- + -|
\2/ \2/ \2 2/
---------------------------------------------
/ 2/x\\ / 2/y\\ / 2/x y\\
|1 + tan |-||*|1 + tan |-||*|1 + tan |- + -||
\ \2// \ \2// \ \2 2//
$$\frac{8 \tan{\left(\frac{x}{2} \right)} \tan{\left(\frac{y}{2} \right)} \tan{\left(\frac{x}{2} + \frac{y}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{y}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{y}{2} \right)} + 1\right)}$$
/x\ /y\ /x y\
8*cot|-|*cot|-|*cot|- + -|
\2/ \2/ \2 2/
---------------------------------------------
/ 2/x\\ / 2/y\\ / 2/x y\\
|1 + cot |-||*|1 + cot |-||*|1 + cot |- + -||
\ \2// \ \2// \ \2 2//
$$\frac{8 \cot{\left(\frac{x}{2} \right)} \cot{\left(\frac{y}{2} \right)} \cot{\left(\frac{x}{2} + \frac{y}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{y}{2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} + \frac{y}{2} \right)} + 1\right)}$$
8 ---------------------------------------------------------------------- / 1 \ / 1 \ / 1 \ /x\ /y\ /x y\ |1 + -------|*|1 + -------|*|1 + -----------|*cot|-|*cot|-|*cot|- + -| | 2/x\| | 2/y\| | 2/x y\| \2/ \2/ \2 2/ | cot |-|| | cot |-|| | cot |- + -|| \ \2// \ \2// \ \2 2//
$$\frac{8}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right) \left(1 + \frac{1}{\cot^{2}{\left(\frac{y}{2} \right)}}\right) \left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} + \frac{y}{2} \right)}}\right) \cot{\left(\frac{x}{2} \right)} \cot{\left(\frac{y}{2} \right)} \cot{\left(\frac{x}{2} + \frac{y}{2} \right)}}$$
/x\ /y\ /x + y\
(1 + cos(x))*(1 + cos(y))*(1 + cos(x + y))*sin|-|*sin|-|*sin|-----|
\2/ \2/ \ 2 /
-------------------------------------------------------------------
/x\ /y\ /x + y\
cos|-|*cos|-|*cos|-----|
\2/ \2/ \ 2 /
$$\frac{\left(\cos{\left(x \right)} + 1\right) \left(\cos{\left(y \right)} + 1\right) \left(\cos{\left(x + y \right)} + 1\right) \sin{\left(\frac{x}{2} \right)} \sin{\left(\frac{y}{2} \right)} \sin{\left(\frac{x + y}{2} \right)}}{\cos{\left(\frac{x}{2} \right)} \cos{\left(\frac{y}{2} \right)} \cos{\left(\frac{x + y}{2} \right)}}$$
/ /x\ /y\\ /x\ /y\ /x\ /y\
4*(1 + cos(x + y))*|tan|-| + tan|-||*cos|-|*cos|-|*sin|-|*sin|-|
\ \2/ \2// \2/ \2/ \2/ \2/
----------------------------------------------------------------
/x\ /y\
1 - tan|-|*tan|-|
\2/ \2/
$$\frac{4 \left(\cos{\left(x + y \right)} + 1\right) \left(\tan{\left(\frac{x}{2} \right)} + \tan{\left(\frac{y}{2} \right)}\right) \sin{\left(\frac{x}{2} \right)} \sin{\left(\frac{y}{2} \right)} \cos{\left(\frac{x}{2} \right)} \cos{\left(\frac{y}{2} \right)}}{- \tan{\left(\frac{x}{2} \right)} \tan{\left(\frac{y}{2} \right)} + 1}$$
/ 2/x pi\\ / 2/y pi\\ / 2/x y pi\\
|1 - cot |- + --||*|1 - cot |- + --||*|1 - cot |- + - + --||*(1 + sin(x))*(1 + sin(y))*(1 + sin(x + y))
\ \2 4 // \ \2 4 // \ \2 2 4 //
-------------------------------------------------------------------------------------------------------
8
$$\frac{\left(- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right) \left(- \cot^{2}{\left(\frac{y}{2} + \frac{\pi}{4} \right)} + 1\right) \left(- \cot^{2}{\left(\frac{x}{2} + \frac{y}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(x \right)} + 1\right) \left(\sin{\left(y \right)} + 1\right) \left(\sin{\left(x + y \right)} + 1\right)}{8}$$
2/x\ 2/y\ 2/x\ 2/y\ 2/x + y\
64*cos |-|*cos |-|*sin |-|*sin |-|*sin |-----|
\2/ \2/ \2/ \2/ \ 2 /
----------------------------------------------
/ 4/x + y\\
| 4*sin |-----||
| \ 2 /|
|1 + -------------|*sin(x)*sin(y)*sin(x + y)
| 2 |
\ sin (x + y) /
$$\frac{64 \sin^{2}{\left(\frac{x}{2} \right)} \sin^{2}{\left(\frac{y}{2} \right)} \sin^{2}{\left(\frac{x + y}{2} \right)} \cos^{2}{\left(\frac{x}{2} \right)} \cos^{2}{\left(\frac{y}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{x + y}{2} \right)}}{\sin^{2}{\left(x + y \right)}} + 1\right) \sin{\left(x \right)} \sin{\left(y \right)} \sin{\left(x + y \right)}}$$
2 2
/ 2/x\\ / 2/y\\ 4/x\ 4/y\ /x\ /y\ /x y\
8*|1 - tan |-|| *|1 - tan |-|| *cos |-|*cos |-|*tan|-|*tan|-|*tan|- + -|
\ \4// \ \4// \4/ \4/ \2/ \2/ \2 2/
------------------------------------------------------------------------
2/x y\
1 + tan |- + -|
\2 2/
$$\frac{8 \left(- \tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2} \left(- \tan^{2}{\left(\frac{y}{4} \right)} + 1\right)^{2} \cos^{4}{\left(\frac{x}{4} \right)} \cos^{4}{\left(\frac{y}{4} \right)} \tan{\left(\frac{x}{2} \right)} \tan{\left(\frac{y}{2} \right)} \tan{\left(\frac{x}{2} + \frac{y}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} + \frac{y}{2} \right)} + 1}$$
// 0 for x mod pi = 0\ // 0 for y mod pi = 0\ // 0 for (x + y) mod pi = 0\ |< |*|< |*|< | \\sin(x) otherwise / \\sin(y) otherwise / \\sin(x + y) otherwise /
$$\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\sin{\left(y \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(x + y\right) \bmod \pi = 0 \\\sin{\left(x + y \right)} & \text{otherwise} \end{cases}\right)$$
// 0 for x mod pi = 0\ // 0 for y mod pi = 0\ // 0 for (x + y) mod pi = 0\ || | || | || | |< 1 |*|< 1 |*|< 1 | ||------ otherwise | ||------ otherwise | ||---------- otherwise | \\csc(x) / \\csc(y) / \\csc(x + y) /
$$\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\csc{\left(x \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\frac{1}{\csc{\left(y \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(x + y\right) \bmod \pi = 0 \\\frac{1}{\csc{\left(x + y \right)}} & \text{otherwise} \end{cases}\right)$$
2/x\ 2/y\ 2/pi x\ 2/pi y\ 2/x y\
64*sin |-|*sin |-|*sin |-- + -|*sin |-- + -|*sin |- + -|
\2/ \2/ \2 2/ \2 2/ \2 2/
--------------------------------------------------------
/ 4/x y\\
| 4*sin |- + -||
| \2 2/|
|1 + -------------|*sin(x)*sin(y)*sin(x + y)
| 2 |
\ sin (x + y) /
$$\frac{64 \sin^{2}{\left(\frac{x}{2} \right)} \sin^{2}{\left(\frac{y}{2} \right)} \sin^{2}{\left(\frac{x}{2} + \frac{\pi}{2} \right)} \sin^{2}{\left(\frac{x}{2} + \frac{y}{2} \right)} \sin^{2}{\left(\frac{y}{2} + \frac{\pi}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{x}{2} + \frac{y}{2} \right)}}{\sin^{2}{\left(x + y \right)}} + 1\right) \sin{\left(x \right)} \sin{\left(y \right)} \sin{\left(x + y \right)}}$$
// 0 for x mod pi = 0\ // 0 for y mod pi = 0\ // 0 for (x + y) mod pi = 0\ || | || | || | |< / pi\ |*|< / pi\ |*|< / pi\ | ||cos|x - --| otherwise | ||cos|y - --| otherwise | ||cos|x + y - --| otherwise | \\ \ 2 / / \\ \ 2 / / \\ \ 2 / /
$$\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\cos{\left(x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\cos{\left(y - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(x + y\right) \bmod \pi = 0 \\\cos{\left(x + y - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)$$
/x\ /y\ /x pi\ /y pi\ /x y pi\
8*cos|-|*cos|-|*cos|- - --|*cos|- - --|*cos|- + - - --|
\2/ \2/ \2 2 / \2 2 / \2 2 2 /
-------------------------------------------------------
/ 2/x y pi\\
| cos |- + - - --||
| \2 2 2 /| /x y\
|1 + ----------------|*cos|- + -|
| 2/x y\ | \2 2/
| cos |- + -| |
\ \2 2/ /
$$\frac{8 \cos{\left(\frac{x}{2} \right)} \cos{\left(\frac{y}{2} \right)} \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)} \cos{\left(\frac{y}{2} - \frac{\pi}{2} \right)} \cos{\left(\frac{x}{2} + \frac{y}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} + \frac{y}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} + \frac{y}{2} \right)}}\right) \cos{\left(\frac{x}{2} + \frac{y}{2} \right)}}$$
2 2
/ 2/x\\ / 2/y\\ /x\ /y\ /x y\
8*|1 - tan |-|| *|1 - tan |-|| *tan|-|*tan|-|*tan|- + -|
\ \4// \ \4// \2/ \2/ \2 2/
--------------------------------------------------------
2 2
/ 2/x\\ / 2/y\\ / 2/x y\\
|1 + tan |-|| *|1 + tan |-|| *|1 + tan |- + -||
\ \4// \ \4// \ \2 2//
$$\frac{8 \left(- \tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2} \left(- \tan^{2}{\left(\frac{y}{4} \right)} + 1\right)^{2} \tan{\left(\frac{x}{2} \right)} \tan{\left(\frac{y}{2} \right)} \tan{\left(\frac{x}{2} + \frac{y}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2} \left(\tan^{2}{\left(\frac{y}{4} \right)} + 1\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} + \frac{y}{2} \right)} + 1\right)}$$
/x y\
8*sec|- + -|
\2 2/
----------------------------------------------------------------------------
/ 2/x y\ \
| sec |- + -| |
| \2 2/ | /x\ /y\ /x pi\ /y pi\ /x y pi\
|1 + ----------------|*sec|-|*sec|-|*sec|- - --|*sec|- - --|*sec|- + - - --|
| 2/x y pi\| \2/ \2/ \2 2 / \2 2 / \2 2 2 /
| sec |- + - - --||
\ \2 2 2 //
$$\frac{8 \sec{\left(\frac{x}{2} + \frac{y}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{x}{2} + \frac{y}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} + \frac{y}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{x}{2} \right)} \sec{\left(\frac{y}{2} \right)} \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)} \sec{\left(\frac{y}{2} - \frac{\pi}{2} \right)} \sec{\left(\frac{x}{2} + \frac{y}{2} - \frac{\pi}{2} \right)}}$$
// 0 for (x + y) mod pi = 0\
|| |
// 0 for x mod pi = 0\ // 0 for y mod pi = 0\ ||1 - cos(x + y) |
|< |*|< |*|<-------------- otherwise |
\\sin(x) otherwise / \\sin(y) otherwise / || /x y\ |
|| tan|- + -| |
\\ \2 2/ /
$$\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\sin{\left(y \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(x + y\right) \bmod \pi = 0 \\\frac{- \cos{\left(x + y \right)} + 1}{\tan{\left(\frac{x}{2} + \frac{y}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
/pi x y\
8*csc|-- - - - -|
\2 2 2/
-----------------------------------------------------------------------
/ 2/pi x y\\
| csc |-- - - - -||
| \2 2 2/| /x\ /y\ /pi x\ /pi y\ /x y\
|1 + ----------------|*csc|-|*csc|-|*csc|-- - -|*csc|-- - -|*csc|- + -|
| 2/x y\ | \2/ \2/ \2 2/ \2 2/ \2 2/
| csc |- + -| |
\ \2 2/ /
$$\frac{8 \csc{\left(- \frac{x}{2} - \frac{y}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} - \frac{y}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} + \frac{y}{2} \right)}}\right) \csc{\left(\frac{x}{2} \right)} \csc{\left(\frac{y}{2} \right)} \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)} \csc{\left(\frac{x}{2} + \frac{y}{2} \right)} \csc{\left(- \frac{y}{2} + \frac{\pi}{2} \right)}}$$
// 0 for x mod pi = 0\ // 0 for y mod pi = 0\ // 0 for (x + y) mod pi = 0\ || | || | || | || 1 | || 1 | || 1 | |<----------- otherwise |*|<----------- otherwise |*|<--------------- otherwise | || / pi\ | || / pi\ | || / pi\ | ||sec|x - --| | ||sec|y - --| | ||sec|x + y - --| | \\ \ 2 / / \\ \ 2 / / \\ \ 2 / /
$$\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\frac{1}{\sec{\left(y - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(x + y\right) \bmod \pi = 0 \\\frac{1}{\sec{\left(x + y - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
/ 2/x pi\\ / 2/y pi\\ / 2/x y pi\\ |-1 + tan |- + --||*|-1 + tan |- + --||*|-1 + tan |- + - + --|| \ \2 4 // \ \2 4 // \ \2 2 4 // --------------------------------------------------------------- / 2/x pi\\ / 2/y pi\\ / 2/x y pi\\ |1 + tan |- + --||*|1 + tan |- + --||*|1 + tan |- + - + --|| \ \2 4 // \ \2 4 // \ \2 2 4 //
$$\frac{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right) \left(\tan^{2}{\left(\frac{y}{2} + \frac{\pi}{4} \right)} - 1\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{y}{2} + \frac{\pi}{4} \right)} - 1\right)}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\tan^{2}{\left(\frac{y}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{y}{2} + \frac{\pi}{4} \right)} + 1\right)}$$
// 0 for (x + y) mod pi = 0\
|| |
|| /x y\ |
// 0 for x mod pi = 0\ // 0 for y mod pi = 0\ || 2*tan|- + -| |
|< |*|< |*|< \2 2/ |
\\sin(x) otherwise / \\sin(y) otherwise / ||--------------- otherwise |
|| 2/x y\ |
||1 + tan |- + -| |
\\ \2 2/ /
$$\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\sin{\left(y \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(x + y\right) \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{x}{2} + \frac{y}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} + \frac{y}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// x \ // y \
|| 1 for - mod 2*pi = 0| || 1 for - mod 2*pi = 0|
|| 2 | || 2 | /x\ /y\ /x y\
8*|< |*|< |*tan|-|*tan|-|*tan|- + -|
||1 + cos(x) | ||1 + cos(y) | \2/ \2/ \2 2/
||---------- otherwise | ||---------- otherwise |
\\ 2 / \\ 2 /
----------------------------------------------------------------------------------------------
2/x y\
1 + tan |- + -|
\2 2/
$$\frac{8 \left(\begin{cases} 1 & \text{for}\: \frac{x}{2} \bmod 2 \pi = 0 \\\frac{\cos{\left(x \right)} + 1}{2} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \frac{y}{2} \bmod 2 \pi = 0 \\\frac{\cos{\left(y \right)} + 1}{2} & \text{otherwise} \end{cases}\right) \tan{\left(\frac{x}{2} \right)} \tan{\left(\frac{y}{2} \right)} \tan{\left(\frac{x}{2} + \frac{y}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} + \frac{y}{2} \right)} + 1}$$
/ 2/x pi\\ / 2/y pi\\ / 2/x y pi\\ |1 - cot |- + --||*|1 - cot |- + --||*|1 - cot |- + - + --|| \ \2 4 // \ \2 4 // \ \2 2 4 // ------------------------------------------------------------ / 2/x pi\\ / 2/y pi\\ / 2/x y pi\\ |1 + cot |- + --||*|1 + cot |- + --||*|1 + cot |- + - + --|| \ \2 4 // \ \2 4 // \ \2 2 4 //
$$\frac{\left(- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right) \left(- \cot^{2}{\left(\frac{y}{2} + \frac{\pi}{4} \right)} + 1\right) \left(- \cot^{2}{\left(\frac{x}{2} + \frac{y}{2} + \frac{\pi}{4} \right)} + 1\right)}{\left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(\frac{y}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} + \frac{y}{2} + \frac{\pi}{4} \right)} + 1\right)}$$
2/x\ 2/y\ 2/x + y\
64*sin |-|*sin |-|*sin |-----|
\2/ \2/ \ 2 /
----------------------------------------------------------------------------
/ 4/x\\ / 4/y\\ / 4/x + y\\
| 4*sin |-|| | 4*sin |-|| | 4*sin |-----||
| \2/| | \2/| | \ 2 /|
|1 + ---------|*|1 + ---------|*|1 + -------------|*sin(x)*sin(y)*sin(x + y)
| 2 | | 2 | | 2 |
\ sin (x) / \ sin (y) / \ sin (x + y) /
$$\frac{64 \sin^{2}{\left(\frac{x}{2} \right)} \sin^{2}{\left(\frac{y}{2} \right)} \sin^{2}{\left(\frac{x + y}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right) \left(\frac{4 \sin^{4}{\left(\frac{y}{2} \right)}}{\sin^{2}{\left(y \right)}} + 1\right) \left(\frac{4 \sin^{4}{\left(\frac{x + y}{2} \right)}}{\sin^{2}{\left(x + y \right)}} + 1\right) \sin{\left(x \right)} \sin{\left(y \right)} \sin{\left(x + y \right)}}$$
// 0 for (x + y) mod pi = 0\
|| |
|| 2 |
// 0 for x mod pi = 0\ // 0 for y mod pi = 0\ ||---------------------------- otherwise |
|< |*|< |*|
$$\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\sin{\left(y \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(x + y\right) \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} + \frac{y}{2} \right)}}\right) \tan{\left(\frac{x}{2} + \frac{y}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
2/x\ 2/y\ 2/x y\
64*sin |-|*sin |-|*sin |- + -|
\2/ \2/ \2 2/
----------------------------------------------------------------------------
/ 4/x\\ / 4/y\\ / 4/x y\\
| 4*sin |-|| | 4*sin |-|| | 4*sin |- + -||
| \2/| | \2/| | \2 2/|
|1 + ---------|*|1 + ---------|*|1 + -------------|*sin(x)*sin(y)*sin(x + y)
| 2 | | 2 | | 2 |
\ sin (x) / \ sin (y) / \ sin (x + y) /
$$\frac{64 \sin^{2}{\left(\frac{x}{2} \right)} \sin^{2}{\left(\frac{y}{2} \right)} \sin^{2}{\left(\frac{x}{2} + \frac{y}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right) \left(\frac{4 \sin^{4}{\left(\frac{y}{2} \right)}}{\sin^{2}{\left(y \right)}} + 1\right) \left(\frac{4 \sin^{4}{\left(\frac{x}{2} + \frac{y}{2} \right)}}{\sin^{2}{\left(x + y \right)}} + 1\right) \sin{\left(x \right)} \sin{\left(y \right)} \sin{\left(x + y \right)}}$$
// 0 for x mod pi = 0\ // 0 for y mod pi = 0\ // 0 for (x + y) mod pi = 0\ || | || | || | ||1 - cos(x) | ||1 - cos(y) | ||1 - cos(x + y) | |<---------- otherwise |*|<---------- otherwise |*|<-------------- otherwise | || /x\ | || /y\ | || /x y\ | || tan|-| | || tan|-| | || tan|- + -| | \\ \2/ / \\ \2/ / \\ \2 2/ /
$$\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{- \cos{\left(x \right)} + 1}{\tan{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\frac{- \cos{\left(y \right)} + 1}{\tan{\left(\frac{y}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(x + y\right) \bmod \pi = 0 \\\frac{- \cos{\left(x + y \right)} + 1}{\tan{\left(\frac{x}{2} + \frac{y}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// 0 for x mod pi = 0\ // 0 for y mod pi = 0\ // 0 for (x + y) mod pi = 0\ || | || | || | || /x\ | || /y\ | || /x y\ | || 2*tan|-| | || 2*tan|-| | || 2*tan|- + -| | |< \2/ |*|< \2/ |*|< \2 2/ | ||----------- otherwise | ||----------- otherwise | ||--------------- otherwise | || 2/x\ | || 2/y\ | || 2/x y\ | ||1 + tan |-| | ||1 + tan |-| | ||1 + tan |- + -| | \\ \2/ / \\ \2/ / \\ \2 2/ /
$$\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{y}{2} \right)}}{\tan^{2}{\left(\frac{y}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(x + y\right) \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{x}{2} + \frac{y}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} + \frac{y}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// 0 for x mod pi = 0\ // 0 for y mod pi = 0\ // 0 for (x + y) mod pi = 0\ || | || | || | || /x\ | || /y\ | || /x y\ | || 2*cot|-| | || 2*cot|-| | || 2*cot|- + -| | |< \2/ |*|< \2/ |*|< \2 2/ | ||----------- otherwise | ||----------- otherwise | ||--------------- otherwise | || 2/x\ | || 2/y\ | || 2/x y\ | ||1 + cot |-| | ||1 + cot |-| | ||1 + cot |- + -| | \\ \2/ / \\ \2/ / \\ \2 2/ /
$$\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{y}{2} \right)}}{\cot^{2}{\left(\frac{y}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(x + y\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} + \frac{y}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} + \frac{y}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// / 3*pi\ \
|| 1 for |x + y + ----| mod 2*pi = 0|
|| \ 2 / |
// / 3*pi\ \ // / 3*pi\ \ || |
|| 1 for |x + ----| mod 2*pi = 0| || 1 for |y + ----| mod 2*pi = 0| || 2/x y pi\ |
|< \ 2 / |*|< \ 2 / |*|<-1 + tan |- + - + --| |
|| | || | || \2 2 4 / |
\\sin(x) otherwise / \\sin(y) otherwise / ||--------------------- otherwise |
|| 2/x y pi\ |
|| 1 + tan |- + - + --| |
\\ \2 2 4 / /
$$\left(\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(y + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(y \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(x + y + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{x}{2} + \frac{y}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{x}{2} + \frac{y}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// 0 for x mod pi = 0\ // 0 for y mod pi = 0\ // 0 for (x + y) mod pi = 0\
|| | || | || |
|
$$\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\sin{\left(y \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(x + y\right) \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \left(x + y\right) \bmod \pi = 0 \\\sin{\left(x + y \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
// 0 for x mod pi = 0\ // 0 for y mod pi = 0\ // 0 for (x + y) mod pi = 0\
|| | || | || |
|| 2 | || 2 | || 2 |
||-------------------- otherwise | ||-------------------- otherwise | ||---------------------------- otherwise |
|
$$\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right) \tan{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{y}{2} \right)}}\right) \tan{\left(\frac{y}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(x + y\right) \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} + \frac{y}{2} \right)}}\right) \tan{\left(\frac{x}{2} + \frac{y}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
/x pi\ /y pi\ /x y pi\
8*cos|- - --|*cos|- - --|*cos|- + - - --|
\2 2 / \2 2 / \2 2 2 /
-------------------------------------------------------------------------------------
/ 2/x pi\\ / 2/y pi\\ / 2/x y pi\\
| cos |- - --|| | cos |- - --|| | cos |- + - - --||
| \2 2 /| | \2 2 /| | \2 2 2 /| /x\ /y\ /x y\
|1 + ------------|*|1 + ------------|*|1 + ----------------|*cos|-|*cos|-|*cos|- + -|
| 2/x\ | | 2/y\ | | 2/x y\ | \2/ \2/ \2 2/
| cos |-| | | cos |-| | | cos |- + -| |
\ \2/ / \ \2/ / \ \2 2/ /
$$\frac{8 \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)} \cos{\left(\frac{y}{2} - \frac{\pi}{2} \right)} \cos{\left(\frac{x}{2} + \frac{y}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right) \left(1 + \frac{\cos^{2}{\left(\frac{y}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{y}{2} \right)}}\right) \left(1 + \frac{\cos^{2}{\left(\frac{x}{2} + \frac{y}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} + \frac{y}{2} \right)}}\right) \cos{\left(\frac{x}{2} \right)} \cos{\left(\frac{y}{2} \right)} \cos{\left(\frac{x}{2} + \frac{y}{2} \right)}}$$
/x\ /y\ /x y\
8*sec|-|*sec|-|*sec|- + -|
\2/ \2/ \2 2/
----------------------------------------------------------------------------------------------------
/ 2/x\ \ / 2/y\ \ / 2/x y\ \
| sec |-| | | sec |-| | | sec |- + -| |
| \2/ | | \2/ | | \2 2/ | /x pi\ /y pi\ /x y pi\
|1 + ------------|*|1 + ------------|*|1 + ----------------|*sec|- - --|*sec|- - --|*sec|- + - - --|
| 2/x pi\| | 2/y pi\| | 2/x y pi\| \2 2 / \2 2 / \2 2 2 /
| sec |- - --|| | sec |- - --|| | sec |- + - - --||
\ \2 2 // \ \2 2 // \ \2 2 2 //
$$\frac{8 \sec{\left(\frac{x}{2} \right)} \sec{\left(\frac{y}{2} \right)} \sec{\left(\frac{x}{2} + \frac{y}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right) \left(\frac{\sec^{2}{\left(\frac{y}{2} \right)}}{\sec^{2}{\left(\frac{y}{2} - \frac{\pi}{2} \right)}} + 1\right) \left(\frac{\sec^{2}{\left(\frac{x}{2} + \frac{y}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} + \frac{y}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)} \sec{\left(\frac{y}{2} - \frac{\pi}{2} \right)} \sec{\left(\frac{x}{2} + \frac{y}{2} - \frac{\pi}{2} \right)}}$$
// x \ // y \
|| 1 for - mod 2*pi = 0| || 1 for - mod 2*pi = 0|
|| 2 | || 2 |
|| | || |
|| 2 | || 2 |
||/ 2/x\\ | ||/ 2/y\\ |
8*|<|-1 + cot |-|| |*|<|-1 + cot |-|| |
||\ \4// | ||\ \4// |
||--------------- otherwise | ||--------------- otherwise |
|| 2 | || 2 |
|| / 2/x\\ | || / 2/y\\ |
|| |1 + cot |-|| | || |1 + cot |-|| |
\\ \ \4// / \\ \ \4// /
-------------------------------------------------------------------------------
/ 1 \ /x\ /y\ /x y\
|1 + -----------|*cot|-|*cot|-|*cot|- + -|
| 2/x y\| \2/ \2/ \2 2/
| cot |- + -||
\ \2 2//
$$\frac{8 \left(\begin{cases} 1 & \text{for}\: \frac{x}{2} \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \frac{y}{2} \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{y}{4} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{y}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} + \frac{y}{2} \right)}}\right) \cot{\left(\frac{x}{2} \right)} \cot{\left(\frac{y}{2} \right)} \cot{\left(\frac{x}{2} + \frac{y}{2} \right)}}$$
/pi x\ /pi y\ /pi x y\
8*csc|-- - -|*csc|-- - -|*csc|-- - - - -|
\2 2/ \2 2/ \2 2 2/
-------------------------------------------------------------------------------------
/ 2/pi x\\ / 2/pi y\\ / 2/pi x y\\
| csc |-- - -|| | csc |-- - -|| | csc |-- - - - -||
| \2 2/| | \2 2/| | \2 2 2/| /x\ /y\ /x y\
|1 + ------------|*|1 + ------------|*|1 + ----------------|*csc|-|*csc|-|*csc|- + -|
| 2/x\ | | 2/y\ | | 2/x y\ | \2/ \2/ \2 2/
| csc |-| | | csc |-| | | csc |- + -| |
\ \2/ / \ \2/ / \ \2 2/ /
$$\frac{8 \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)} \csc{\left(- \frac{y}{2} + \frac{\pi}{2} \right)} \csc{\left(- \frac{x}{2} - \frac{y}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right) \left(1 + \frac{\csc^{2}{\left(- \frac{y}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{y}{2} \right)}}\right) \left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} - \frac{y}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} + \frac{y}{2} \right)}}\right) \csc{\left(\frac{x}{2} \right)} \csc{\left(\frac{y}{2} \right)} \csc{\left(\frac{x}{2} + \frac{y}{2} \right)}}$$
// / 3*pi\ \ // / 3*pi\ \ // / 3*pi\ \ || 1 for |x + ----| mod 2*pi = 0| || 1 for |y + ----| mod 2*pi = 0| || 1 for |x + y + ----| mod 2*pi = 0| || \ 2 / | || \ 2 / | || \ 2 / | || | || | || | || 2/x pi\ | || 2/y pi\ | || 2/x y pi\ | |<-1 + tan |- + --| |*|<-1 + tan |- + --| |*|<-1 + tan |- + - + --| | || \2 4 / | || \2 4 / | || \2 2 4 / | ||----------------- otherwise | ||----------------- otherwise | ||--------------------- otherwise | || 2/x pi\ | || 2/y pi\ | || 2/x y pi\ | || 1 + tan |- + --| | || 1 + tan |- + --| | || 1 + tan |- + - + --| | \\ \2 4 / / \\ \2 4 / / \\ \2 2 4 / /
$$\left(\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(y + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{y}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{y}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(x + y + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{x}{2} + \frac{y}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{x}{2} + \frac{y}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// 0 for x mod pi = 0\ // 0 for y mod pi = 0\ // 0 for (x + y) mod pi = 0\ || | || | || | || sin(x) | || sin(y) | || sin(x + y) | ||----------------------- otherwise | ||----------------------- otherwise | ||------------------------------- otherwise | ||/ 2 \ | ||/ 2 \ | ||/ 2 \ | |<| sin (x) | 2/x\ |*|<| sin (y) | 2/y\ |*|<| sin (x + y) | 2/x y\ | |||1 + ---------|*sin |-| | |||1 + ---------|*sin |-| | |||1 + -------------|*sin |- + -| | ||| 4/x\| \2/ | ||| 4/y\| \2/ | ||| 4/x y\| \2 2/ | ||| 4*sin |-|| | ||| 4*sin |-|| | ||| 4*sin |- + -|| | ||\ \2// | ||\ \2// | ||\ \2 2// | \\ / \\ / \\ /
$$\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{\sin{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right) \sin^{2}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\frac{\sin{\left(y \right)}}{\left(1 + \frac{\sin^{2}{\left(y \right)}}{4 \sin^{4}{\left(\frac{y}{2} \right)}}\right) \sin^{2}{\left(\frac{y}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(x + y\right) \bmod \pi = 0 \\\frac{\sin{\left(x + y \right)}}{\left(1 + \frac{\sin^{2}{\left(x + y \right)}}{4 \sin^{4}{\left(\frac{x}{2} + \frac{y}{2} \right)}}\right) \sin^{2}{\left(\frac{x}{2} + \frac{y}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// 0 for x mod pi = 0\ // 0 for y mod pi = 0\ // 0 for (x + y) mod pi = 0\ || | || | || | || 2*sin(x) | || 2*sin(y) | || 2*sin(x + y) | ||---------------------------- otherwise | ||---------------------------- otherwise | ||------------------------------------ otherwise | || / 2 \ | || / 2 \ | || / 2 \ | |< | sin (x) | |*|< | sin (y) | |*|< | sin (x + y) | | ||(1 - cos(x))*|1 + ---------| | ||(1 - cos(y))*|1 + ---------| | ||(1 - cos(x + y))*|1 + -------------| | || | 4/x\| | || | 4/y\| | || | 4/x + y\| | || | 4*sin |-|| | || | 4*sin |-|| | || | 4*sin |-----|| | || \ \2// | || \ \2// | || \ \ 2 // | \\ / \\ / \\ /
$$\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \sin{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right) \left(- \cos{\left(x \right)} + 1\right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\frac{2 \sin{\left(y \right)}}{\left(1 + \frac{\sin^{2}{\left(y \right)}}{4 \sin^{4}{\left(\frac{y}{2} \right)}}\right) \left(- \cos{\left(y \right)} + 1\right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(x + y\right) \bmod \pi = 0 \\\frac{2 \sin{\left(x + y \right)}}{\left(1 + \frac{\sin^{2}{\left(x + y \right)}}{4 \sin^{4}{\left(\frac{x + y}{2} \right)}}\right) \left(- \cos{\left(x + y \right)} + 1\right)} & \text{otherwise} \end{cases}\right)$$
// 0 for x mod pi = 0\ // 0 for y mod pi = 0\ // 0 for (x + y) mod pi = 0\ || | || | || | ||/ 0 for x mod pi = 0 | ||/ 0 for y mod pi = 0 | ||/ 0 for (x + y) mod pi = 0 | ||| | ||| | ||| | ||| /x\ | ||| /y\ | ||| /x y\ | |<| 2*cot|-| |*|<| 2*cot|-| |*|<| 2*cot|- + -| | ||< \2/ otherwise | ||< \2/ otherwise | ||< \2 2/ otherwise | |||----------- otherwise | |||----------- otherwise | |||--------------- otherwise | ||| 2/x\ | ||| 2/y\ | ||| 2/x y\ | |||1 + cot |-| | |||1 + cot |-| | |||1 + cot |- + -| | \\\ \2/ / \\\ \2/ / \\\ \2 2/ /
$$\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{y}{2} \right)}}{\cot^{2}{\left(\frac{y}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(x + y\right) \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \left(x + y\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} + \frac{y}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} + \frac{y}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
// 0 for x mod pi = 0\ // 0 for y mod pi = 0\ // 0 for (x + y) mod pi = 0\
|| | || | || |
|| /x pi\ | || /y pi\ | || /x y pi\ |
|| 2*sec|- - --| | || 2*sec|- - --| | || 2*sec|- + - - --| |
|| \2 2 / | || \2 2 / | || \2 2 2 / |
||------------------------- otherwise | ||------------------------- otherwise | ||--------------------------------- otherwise |
|
$$\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right) \sec{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{y}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{y}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{y}{2} \right)}}\right) \sec{\left(\frac{y}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(x + y\right) \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{x}{2} + \frac{y}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} + \frac{y}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} + \frac{y}{2} \right)}}\right) \sec{\left(\frac{x}{2} + \frac{y}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// 0 for x mod pi = 0\ // 0 for y mod pi = 0\ // 0 for (x + y) mod pi = 0\
|| | || | || |
|| /x\ | || /y\ | || /x y\ |
|| 2*cos|-| | || 2*cos|-| | || 2*cos|- + -| |
|| \2/ | || \2/ | || \2 2/ |
||------------------------------ otherwise | ||------------------------------ otherwise | ||-------------------------------------- otherwise |
|
$$\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{x}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{y}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{y}{2} \right)}}{\cos^{2}{\left(\frac{y}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{y}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(x + y\right) \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{x}{2} + \frac{y}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{x}{2} + \frac{y}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} + \frac{y}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{x}{2} + \frac{y}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// 0 for x mod pi = 0\ // 0 for y mod pi = 0\ // 0 for (x + y) mod pi = 0\
|| | || | || |
|| /x\ | || /y\ | || /x y\ |
|| 2*csc|-| | || 2*csc|-| | || 2*csc|- + -| |
|| \2/ | || \2/ | || \2 2/ |
||------------------------------ otherwise | ||------------------------------ otherwise | ||-------------------------------------- otherwise |
|
$$\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{x}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{y}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{y}{2} \right)}}{\csc^{2}{\left(- \frac{y}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{y}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(x + y\right) \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{x}{2} + \frac{y}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} + \frac{y}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} - \frac{y}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{x}{2} - \frac{y}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
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((0, Mod(y = pi, 0)), (2*csc(y/2)/((1 + csc(y/2)^2/csc(pi/2 - y/2)^2)*csc(pi/2 - y/2)), True))*Piecewise((0, Mod(x + y = pi, 0)), (2*csc(x/2 + y/2)/((1 + csc(x/2 + y/2)^2/csc(pi/2 - x/2 - y/2)^2)*csc(pi/2 - x/2 - y/2)), True))
Степени
[src]
/ I*(-x - y) I*(x + y)\ / -I*x I*x\ / -I*y I*y\
I*\- e + e /*\- e + e /*\- e + e /
----------------------------------------------------------------
8
$$\frac{i \left(e^{i x} - e^{- i x}\right) \left(e^{i y} - e^{- i y}\right) \left(- e^{i \left(- x - y\right)} + e^{i \left(x + y\right)}\right)}{8}$$
i*(-exp(i*(-x - y)) + exp(i*(x + y)))*(-exp(-i*x) + exp(i*x))*(-exp(-i*y) + exp(i*y))/8