Господин Экзамен
Другие калькуляторы
-
Как пользоваться?
- sin(a+b)-sin(a-b) если a=-1/2
- b*(2*d-5)-b*(d+5) если b=2
- x^4-5*x^3+3*x-15 если x=3
- sqrt(4)+y+1/(sqrt(15*y))-5 если y=-2
- Разложить многочлен на множители:
- a*h^7-a*m-u*h^7+n*m+u*m-n*h^7
- x^6*y^9-64*x^3
- 3*x^2+12*x+13
- x^3-6*x^2+9*x-3
- Общий знаменатель:
- 5/x-7-2/x-3*x/x^2-49+21/49-x^2
- a^2+16/a-4+8*a/4+a
- exp(-x)*sin(x)/2+x*exp(-x)*sin(x)/2-x*cos(x)*exp(-x)/2
- a3/4-b1/2*a1/4+a1/2*b1/4-b3/4
- Умножение столбиком:
- 54*11
- 79*97
- 63*5197
- 7900*35
- Деление столбиком:
- 87100/2931
- 721/3
- 1071/3
- 3996/3
- Раскрыть скобки в:
- x*(-6)*(x+3)
- 10-8*(-3-x)
- (x^2-25)*(x+4)
- 4*x^2+19*x-5
- Разложение числа на простые множители:
- 318
- 2744
- 3668
- 3960
- График функции y =:
- -1/2
- Интеграл d{x}:
-
-1/2
- Производная:
- -1/2
-
Идентичные выражения
- sin(a+b)-sin(a-b) если a=- один / два
- синус от (a плюс b) минус синус от (a минус b) если a равно минус 1 делить на 2
- синус от (a плюс b) минус синус от (a минус b) если a равно минус один делить на два
- sina+b-sina-b если a=-1/2
- sin(a+b)-sin(a-b) при a=-1/2
- sin(a+b)-sin(a-b) если a=-1 разделить на 2
-
Похожие выражения
- sin(a+b)+sin(a-b) если a=-1/2
- sin(a+b)-sin(a+b) если a=-1/2
- sin(a-b)-sin(a-b) если a=-1/2
- (sin(a+b)+sin(a-b))/(sin(a+b)-sin(a-b)) если a=-1
- sin(a+b)-sin(a-b) если a=+1/2
sin(a+b)-sin(a-b) если a=-1/2
Решение
Вы ввели
[src]
sin(a + b) - sin(a - b)
$$- \sin{\left(a - b \right)} + \sin{\left(a + b \right)}$$
sin(a + b) - sin(a - b)
Общее упрощение
[src]
2*cos(a)*sin(b)
$$2 \sin{\left(b \right)} \cos{\left(a \right)}$$
2*cos(a)*sin(b)
Подстановка условия
[src]
sin(a + b) - sin(a - b) при a = -1/2
подставляем
sin(a + b) - sin(a - b)
$$- \sin{\left(a - b \right)} + \sin{\left(a + b \right)}$$
2*cos(a)*sin(b)
$$2 \sin{\left(b \right)} \cos{\left(a \right)}$$
переменные
a = -1/2
$$a = - \frac{1}{2}$$
2*cos((-1/2))*sin(b)
$$2 \sin{\left(b \right)} \cos{\left((-1/2) \right)}$$
2*cos(-1/2)*sin(b)
$$2 \sin{\left(b \right)} \cos{\left(- \frac{1}{2} \right)}$$
2*cos(1/2)*sin(b)
$$2 \sin{\left(b \right)} \cos{\left(\frac{1}{2} \right)}$$
2*cos(1/2)*sin(b)
Степени
[src]
/ I*(b - a) I*(a - b)\ / I*(-a - b) I*(a + b)\
I*\- e + e / I*\- e + e /
----------------------------- - ------------------------------
2 2
$$- \frac{i \left(- e^{i \left(- a - b\right)} + e^{i \left(a + b\right)}\right)}{2} + \frac{i \left(- e^{i \left(- a + b\right)} + e^{i \left(a - b\right)}\right)}{2}$$
i*(-exp(i*(b - a)) + exp(i*(a - b)))/2 - i*(-exp(i*(-a - b)) + exp(i*(a + b)))/2
Тригонометрическая часть
[src]
2*cos(a)*sin(b)
$$2 \sin{\left(b \right)} \cos{\left(a \right)}$$
/ pi\
2*sin(b)*sin|a + --|
\ 2 /
$$2 \sin{\left(b \right)} \sin{\left(a + \frac{\pi}{2} \right)}$$
/ pi\
2*cos(a)*cos|b - --|
\ 2 /
$$2 \cos{\left(a \right)} \cos{\left(b - \frac{\pi}{2} \right)}$$
2
------------------
/ pi\
sec(a)*sec|b - --|
\ 2 /
$$\frac{2}{\sec{\left(a \right)} \sec{\left(b - \frac{\pi}{2} \right)}}$$
2
------------------
/pi \
csc(b)*csc|-- - a|
\2 /
$$\frac{2}{\csc{\left(b \right)} \csc{\left(- a + \frac{\pi}{2} \right)}}$$
1 1 ---------- - ---------- csc(a + b) csc(a - b)
$$\frac{1}{\csc{\left(a + b \right)}} - \frac{1}{\csc{\left(a - b \right)}}$$
/ pi\ / pi\
- cos|a - b - --| + cos|a + b - --|
\ 2 / \ 2 /
$$- \cos{\left(a - b - \frac{\pi}{2} \right)} + \cos{\left(a + b - \frac{\pi}{2} \right)}$$
1 1 --------------- - --------------- csc(pi - a - b) csc(pi + b - a)
$$- \frac{1}{\csc{\left(- a + b + \pi \right)}} + \frac{1}{\csc{\left(- a - b + \pi \right)}}$$
1 1 --------------- - --------------- / pi\ / pi\ sec|a + b - --| sec|a - b - --| \ 2 / \ 2 /
$$\frac{1}{\sec{\left(a + b - \frac{\pi}{2} \right)}} - \frac{1}{\sec{\left(a - b - \frac{\pi}{2} \right)}}$$
1 1 --------------- - --------------- /pi \ / pi \ sec|-- - a - b| sec|b + -- - a| \2 / \ 2 /
$$- \frac{1}{\sec{\left(- a + b + \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(- a - b + \frac{\pi}{2} \right)}}$$
/ 2/a\\ /b\ /b\ 4*|-1 + 2*cos |-||*cos|-|*sin|-| \ \2// \2/ \2/
$$4 \cdot \left(2 \cos^{2}{\left(\frac{a}{2} \right)} - 1\right) \sin{\left(\frac{b}{2} \right)} \cos{\left(\frac{b}{2} \right)}$$
/ 2/a\\ /b\ /b pi\ 4*|-1 + 2*cos |-||*cos|-|*cos|- - --| \ \2// \2/ \2 2 /
$$4 \cdot \left(2 \cos^{2}{\left(\frac{a}{2} \right)} - 1\right) \cos{\left(\frac{b}{2} \right)} \cos{\left(\frac{b}{2} - \frac{\pi}{2} \right)}$$
/ 2 \ 4*|-1 + -------| | 2/a\| | sec |-|| \ \2// ------------------ /b\ /b pi\ sec|-|*sec|- - --| \2/ \2 2 /
$$\frac{4 \left(-1 + \frac{2}{\sec^{2}{\left(\frac{a}{2} \right)}}\right)}{\sec{\left(\frac{b}{2} \right)} \sec{\left(\frac{b}{2} - \frac{\pi}{2} \right)}}$$
/ 2/pi a\\ /b\ /pi b\ 4*|-1 + 2*sin |-- + -||*sin|-|*sin|-- + -| \ \2 2// \2/ \2 2/
$$4 \cdot \left(2 \sin^{2}{\left(\frac{a}{2} + \frac{\pi}{2} \right)} - 1\right) \sin{\left(\frac{b}{2} \right)} \sin{\left(\frac{b}{2} + \frac{\pi}{2} \right)}$$
/ 2 \
4*|-1 + ------------|
| 2/pi a\|
| csc |-- - -||
\ \2 2//
---------------------
/b\ /pi b\
csc|-|*csc|-- - -|
\2/ \2 2/
$$\frac{4 \left(-1 + \frac{2}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}\right)}{\csc{\left(\frac{b}{2} \right)} \csc{\left(- \frac{b}{2} + \frac{\pi}{2} \right)}}$$
/ 2/a\\ /b\
4*|1 - tan |-||*tan|-|
\ \2// \2/
---------------------------
/ 2/a\\ / 2/b\\
|1 + tan |-||*|1 + tan |-||
\ \2// \ \2//
$$\frac{4 \cdot \left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \tan{\left(\frac{b}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{b}{2} \right)} + 1\right)}$$
/a b\ /a b\
2*cot|- - -| 2*cot|- + -|
\2 2/ \2 2/
- --------------- + ---------------
2/a b\ 2/a b\
1 + cot |- - -| 1 + cot |- + -|
\2 2/ \2 2/
$$\frac{2 \cot{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)} + 1} - \frac{2 \cot{\left(\frac{a}{2} - \frac{b}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} + 1}$$
/a b\ /a b\
2*tan|- - -| 2*tan|- + -|
\2 2/ \2 2/
- --------------- + ---------------
2/a b\ 2/a b\
1 + tan |- - -| 1 + tan |- + -|
\2 2/ \2 2/
$$\frac{2 \tan{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)} + 1} - \frac{2 \tan{\left(\frac{a}{2} - \frac{b}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} + 1}$$
// 0 for b mod pi = 0\ // 1 for a mod 2*pi = 0\ 2*|< |*|< | \\sin(b) otherwise / \\cos(a) otherwise /
$$2 \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\sin{\left(b \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right)$$
/a b\ /a b\ /a b\ /a + b\
- tan|- - -| - cos(a - b)*tan|- - -| + (1 + cos(a + b))*sec|- + -|*sin|-----|
\2 2/ \2 2/ \2 2/ \ 2 /
$$\left(\cos{\left(a + b \right)} + 1\right) \sin{\left(\frac{a + b}{2} \right)} \sec{\left(\frac{a}{2} + \frac{b}{2} \right)} - \cos{\left(a - b \right)} \tan{\left(\frac{a}{2} - \frac{b}{2} \right)} - \tan{\left(\frac{a}{2} - \frac{b}{2} \right)}$$
/a + b\ /a - b\
(1 + cos(a + b))*sin|-----| (1 + cos(a - b))*sin|-----|
\ 2 / \ 2 /
--------------------------- - ---------------------------
/a + b\ /a - b\
cos|-----| cos|-----|
\ 2 / \ 2 /
$$- \frac{\left(\cos{\left(a - b \right)} + 1\right) \sin{\left(\frac{a - b}{2} \right)}}{\cos{\left(\frac{a - b}{2} \right)}} + \frac{\left(\cos{\left(a + b \right)} + 1\right) \sin{\left(\frac{a + b}{2} \right)}}{\cos{\left(\frac{a + b}{2} \right)}}$$
2 2 - ---------------------------- + ---------------------------- / 1 \ /a b\ / 1 \ /a b\ |1 + -----------|*cot|- - -| |1 + -----------|*cot|- + -| | 2/a b\| \2 2/ | 2/a b\| \2 2/ | cot |- - -|| | cot |- + -|| \ \2 2// \ \2 2//
$$\frac{2}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}}\right) \cot{\left(\frac{a}{2} + \frac{b}{2} \right)}} - \frac{2}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}\right) \cot{\left(\frac{a}{2} - \frac{b}{2} \right)}}$$
// 0 for (a - b) mod pi = 0\ // 0 for (a + b) mod pi = 0\ - |< | + |< | \\sin(a - b) otherwise / \\sin(a + b) otherwise /
$$\left(- \begin{cases} 0 & \text{for}\: \left(a - b\right) \bmod \pi = 0 \\\sin{\left(a - b \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(a + b\right) \bmod \pi = 0 \\\sin{\left(a + b \right)} & \text{otherwise} \end{cases}\right)$$
2/a b pi\ / 2/b a pi\\
-1 + tan |- + - + --| (1 - sin(a - b))*|-1 + cot |- - - + --||
\2 2 4 / \ \2 2 4 //
--------------------- - ----------------------------------------
2/a b pi\ 2
1 + tan |- + - + --|
\2 2 4 /
$$- \frac{\left(- \sin{\left(a - b \right)} + 1\right) \left(\cot^{2}{\left(- \frac{a}{2} + \frac{b}{2} + \frac{\pi}{4} \right)} - 1\right)}{2} + \frac{\tan^{2}{\left(\frac{a}{2} + \frac{b}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} + \frac{b}{2} + \frac{\pi}{4} \right)} + 1}$$
/ 2/a b pi\\ 2/b a pi\
|1 - cot |- + - + --||*(1 + sin(a + b)) 1 - tan |- - - + --|
\ \2 2 4 // \2 2 4 /
--------------------------------------- - --------------------
2 2/b a pi\
1 + tan |- - - + --|
\2 2 4 /
$$\frac{\left(- \cot^{2}{\left(\frac{a}{2} + \frac{b}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(a + b \right)} + 1\right)}{2} - \frac{- \tan^{2}{\left(- \frac{a}{2} + \frac{b}{2} + \frac{\pi}{4} \right)} + 1}{\tan^{2}{\left(- \frac{a}{2} + \frac{b}{2} + \frac{\pi}{4} \right)} + 1}$$
// 0 for (a - b) mod pi = 0\ // 0 for (a + b) mod pi = 0\ || | || | - |< 1 | + |< 1 | ||---------- otherwise | ||---------- otherwise | \\csc(a - b) / \\csc(a + b) /
$$\left(- \begin{cases} 0 & \text{for}\: \left(a - b\right) \bmod \pi = 0 \\\frac{1}{\csc{\left(a - b \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(a + b\right) \bmod \pi = 0 \\\frac{1}{\csc{\left(a + b \right)}} & \text{otherwise} \end{cases}\right)$$
2/a b pi\ 2/b a pi\
-1 + tan |- + - + --| -1 + cot |- - - + --|
\2 2 4 / \2 2 4 /
--------------------- - ---------------------
2/a b pi\ 2/b a pi\
1 + tan |- + - + --| 1 + cot |- - - + --|
\2 2 4 / \2 2 4 /
$$\frac{\tan^{2}{\left(\frac{a}{2} + \frac{b}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} + \frac{b}{2} + \frac{\pi}{4} \right)} + 1} - \frac{\cot^{2}{\left(- \frac{a}{2} + \frac{b}{2} + \frac{\pi}{4} \right)} - 1}{\cot^{2}{\left(- \frac{a}{2} + \frac{b}{2} + \frac{\pi}{4} \right)} + 1}$$
/ 2\
| / 2/a\\ |
| 2*|1 - tan |-|| |
/ 2/b\\ | \ \4// | /b\
8*|1 - tan |-||*|-1 + ----------------|*tan|-|
\ \4// | 2 | \4/
| / 2/a\\ |
| |1 + tan |-|| |
\ \ \4// /
----------------------------------------------
2
/ 2/b\\
|1 + tan |-||
\ \4//
$$\frac{8 \cdot \left(- \tan^{2}{\left(\frac{b}{4} \right)} + 1\right) \left(\frac{2 \left(- \tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2}} - 1\right) \tan{\left(\frac{b}{4} \right)}}{\left(\tan^{2}{\left(\frac{b}{4} \right)} + 1\right)^{2}}$$
// 0 for (a - b) mod pi = 0\ // 0 for (a + b) mod pi = 0\ || | || | - |< / pi\ | + |< / pi\ | ||cos|a - b - --| otherwise | ||cos|a + b - --| otherwise | \\ \ 2 / / \\ \ 2 / /
$$\left(- \begin{cases} 0 & \text{for}\: \left(a - b\right) \bmod \pi = 0 \\\cos{\left(a - b - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(a + b\right) \bmod \pi = 0 \\\cos{\left(a + b - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)$$
2/a b pi\ 2/b a pi\
1 - cot |- + - + --| 1 - tan |- - - + --|
\2 2 4 / \2 2 4 /
-------------------- - --------------------
2/a b pi\ 2/b a pi\
1 + cot |- + - + --| 1 + tan |- - - + --|
\2 2 4 / \2 2 4 /
$$- \frac{- \tan^{2}{\left(- \frac{a}{2} + \frac{b}{2} + \frac{\pi}{4} \right)} + 1}{\tan^{2}{\left(- \frac{a}{2} + \frac{b}{2} + \frac{\pi}{4} \right)} + 1} + \frac{- \cot^{2}{\left(\frac{a}{2} + \frac{b}{2} + \frac{\pi}{4} \right)} + 1}{\cot^{2}{\left(\frac{a}{2} + \frac{b}{2} + \frac{\pi}{4} \right)} + 1}$$
// 0 for (a - b) mod pi = 0\ // 0 for (a + b) mod pi = 0\ || | || | || 1 | || 1 | - |<--------------- otherwise | + |<--------------- otherwise | || / pi\ | || / pi\ | ||sec|a - b - --| | ||sec|a + b - --| | \\ \ 2 / / \\ \ 2 / /
$$\left(- \begin{cases} 0 & \text{for}\: \left(a - b\right) \bmod \pi = 0 \\\frac{1}{\sec{\left(a - b - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(a + b\right) \bmod \pi = 0 \\\frac{1}{\sec{\left(a + b - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
2/a - b\ 2/a + b\
4*sin |-----| 4*sin |-----|
\ 2 / \ 2 /
- ------------------------------ + ------------------------------
/ 4/a - b\\ / 4/a + b\\
| 4*sin |-----|| | 4*sin |-----||
| \ 2 /| | \ 2 /|
|1 + -------------|*sin(a - b) |1 + -------------|*sin(a + b)
| 2 | | 2 |
\ sin (a - b) / \ sin (a + b) /
$$\frac{4 \sin^{2}{\left(\frac{a + b}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{a + b}{2} \right)}}{\sin^{2}{\left(a + b \right)}} + 1\right) \sin{\left(a + b \right)}} - \frac{4 \sin^{2}{\left(\frac{a - b}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{a - b}{2} \right)}}{\sin^{2}{\left(a - b \right)}} + 1\right) \sin{\left(a - b \right)}}$$
// 0 for b mod pi = 0\ // 1 for a mod 2*pi = 0\ || | || | || /b\ | || 2/a\ | || 2*cot|-| | ||-1 + cot |-| | 2*|< \2/ |*|< \2/ | ||----------- otherwise | ||------------ otherwise | || 2/b\ | || 2/a\ | ||1 + cot |-| | ||1 + cot |-| | \\ \2/ / \\ \2/ /
$$2 \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{b}{2} \right)}}{\cot^{2}{\left(\frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
2/a b\ 2/a b\
4*sin |- - -| 4*sin |- + -|
\2 2/ \2 2/
- ------------------------------ + ------------------------------
/ 4/a b\\ / 4/a b\\
| 4*sin |- - -|| | 4*sin |- + -||
| \2 2/| | \2 2/|
|1 + -------------|*sin(a - b) |1 + -------------|*sin(a + b)
| 2 | | 2 |
\ sin (a - b) / \ sin (a + b) /
$$\frac{4 \sin^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\sin^{2}{\left(a + b \right)}} + 1\right) \sin{\left(a + b \right)}} - \frac{4 \sin^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} - \frac{b}{2} \right)}}{\sin^{2}{\left(a - b \right)}} + 1\right) \sin{\left(a - b \right)}}$$
// 0 for (a - b) mod pi = 0\ // 0 for (a + b) mod pi = 0\ || | || | ||1 - cos(a - b) | ||1 - cos(a + b) | - |<-------------- otherwise | + |<-------------- otherwise | || /a b\ | || /a b\ | || tan|- - -| | || tan|- + -| | \\ \2 2/ / \\ \2 2/ /
$$\left(- \begin{cases} 0 & \text{for}\: \left(a - b\right) \bmod \pi = 0 \\\frac{- \cos{\left(a - b \right)} + 1}{\tan{\left(\frac{a}{2} - \frac{b}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(a + b\right) \bmod \pi = 0 \\\frac{- \cos{\left(a + b \right)} + 1}{\tan{\left(\frac{a}{2} + \frac{b}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// 0 for (a - b) mod pi = 0\ // 0 for (a + b) mod pi = 0\ || | || | || /a b\ | || /a b\ | || 2*tan|- - -| | || 2*tan|- + -| | - |< \2 2/ | + |< \2 2/ | ||--------------- otherwise | ||--------------- otherwise | || 2/a b\ | || 2/a b\ | ||1 + tan |- - -| | ||1 + tan |- + -| | \\ \2 2/ / \\ \2 2/ /
$$\left(- \begin{cases} 0 & \text{for}\: \left(a - b\right) \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{a}{2} - \frac{b}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(a + b\right) \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// 0 for (a - b) mod pi = 0\ // 0 for (a + b) mod pi = 0\ || | || | || /a b\ | || /a b\ | || 2*cot|- - -| | || 2*cot|- + -| | - |< \2 2/ | + |< \2 2/ | ||--------------- otherwise | ||--------------- otherwise | || 2/a b\ | || 2/a b\ | ||1 + cot |- - -| | ||1 + cot |- + -| | \\ \2 2/ / \\ \2 2/ /
$$\left(- \begin{cases} 0 & \text{for}\: \left(a - b\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} - \frac{b}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(a + b\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
/ // a \\ // b \ // b \ | || 1 for - mod 2*pi = 0|| || 0 for - mod pi = 0| || 1 for - mod 2*pi = 0| | || 2 || || 2 | || 2 | 4*|-1 + 2*|< ||*|< |*|< | | ||1 + cos(a) || || /b\ | || /b\ | | ||---------- otherwise || ||sin|-| otherwise | ||cos|-| otherwise | \ \\ 2 // \\ \2/ / \\ \2/ /
$$4 \cdot \left(\left(2 \left(\begin{cases} 1 & \text{for}\: \frac{a}{2} \bmod 2 \pi = 0 \\\frac{\cos{\left(a \right)} + 1}{2} & \text{otherwise} \end{cases}\right)\right) - 1\right) \left(\begin{cases} 0 & \text{for}\: \frac{b}{2} \bmod \pi = 0 \\\sin{\left(\frac{b}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \frac{b}{2} \bmod 2 \pi = 0 \\\cos{\left(\frac{b}{2} \right)} & \text{otherwise} \end{cases}\right)$$
// 0 for (a - b) mod pi = 0\ // 0 for (a + b) mod pi = 0\
|| | || |
- |
$$\left(- \begin{cases} 0 & \text{for}\: \left(a - b\right) \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \left(a - b\right) \bmod \pi = 0 \\\sin{\left(a - b \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(a + b\right) \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \left(a + b\right) \bmod \pi = 0 \\\sin{\left(a + b \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
// 0 for (a - b) mod pi = 0\ // 0 for (a + b) mod pi = 0\
|| | || |
|| 2 | || 2 |
||---------------------------- otherwise | ||---------------------------- otherwise |
- |
$$\left(- \begin{cases} 0 & \text{for}\: \left(a - b\right) \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}\right) \tan{\left(\frac{a}{2} - \frac{b}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(a + b\right) \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}}\right) \tan{\left(\frac{a}{2} + \frac{b}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
/a pi b\ /a b pi\
2*cos|- - -- - -| 2*cos|- + - - --|
\2 2 2/ \2 2 2 /
- --------------------------------- + ---------------------------------
/ 2/a pi b\\ / 2/a b pi\\
| cos |- - -- - -|| | cos |- + - - --||
| \2 2 2/| /a b\ | \2 2 2 /| /a b\
|1 + ----------------|*cos|- - -| |1 + ----------------|*cos|- + -|
| 2/a b\ | \2 2/ | 2/a b\ | \2 2/
| cos |- - -| | | cos |- + -| |
\ \2 2/ / \ \2 2/ /
$$\frac{2 \cos{\left(\frac{a}{2} + \frac{b}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{a}{2} + \frac{b}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}}\right) \cos{\left(\frac{a}{2} + \frac{b}{2} \right)}} - \frac{2 \cos{\left(\frac{a}{2} - \frac{b}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{b}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}\right) \cos{\left(\frac{a}{2} - \frac{b}{2} \right)}}$$
/a b\ /a b\
2*sec|- - -| 2*sec|- + -|
\2 2/ \2 2/
- -------------------------------------- + --------------------------------------
/ 2/a b\ \ / 2/a b\ \
| sec |- - -| | | sec |- + -| |
| \2 2/ | /a pi b\ | \2 2/ | /a b pi\
|1 + ----------------|*sec|- - -- - -| |1 + ----------------|*sec|- + - - --|
| 2/a pi b\| \2 2 2/ | 2/a b pi\| \2 2 2 /
| sec |- - -- - -|| | sec |- + - - --||
\ \2 2 2// \ \2 2 2 //
$$\frac{2 \sec{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} + \frac{b}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{a}{2} + \frac{b}{2} - \frac{\pi}{2} \right)}} - \frac{2 \sec{\left(\frac{a}{2} - \frac{b}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{b}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{a}{2} - \frac{b}{2} - \frac{\pi}{2} \right)}}$$
/pi b a\ /pi a b\
2*csc|-- + - - -| 2*csc|-- - - - -|
\2 2 2/ \2 2 2/
- --------------------------------- + ---------------------------------
/ 2/pi b a\\ / 2/pi a b\\
| csc |-- + - - -|| | csc |-- - - - -||
| \2 2 2/| /a b\ | \2 2 2/| /a b\
|1 + ----------------|*csc|- - -| |1 + ----------------|*csc|- + -|
| 2/a b\ | \2 2/ | 2/a b\ | \2 2/
| csc |- - -| | | csc |- + -| |
\ \2 2/ / \ \2 2/ /
$$\frac{2 \csc{\left(- \frac{a}{2} - \frac{b}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} - \frac{b}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}}\right) \csc{\left(\frac{a}{2} + \frac{b}{2} \right)}} - \frac{2 \csc{\left(- \frac{a}{2} + \frac{b}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{b}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}\right) \csc{\left(\frac{a}{2} - \frac{b}{2} \right)}}$$
// / 3*pi\ \
// / 3*pi\ \ || 1 for |a + b + ----| mod 2*pi = 0|
|| 1 for |a - b + ----| mod 2*pi = 0| || \ 2 / |
|| \ 2 / | || |
|| | || 2/a b pi\ |
- |< / 2/b a pi\\ | + |<-1 + tan |- + - + --| |
||(1 - sin(a - b))*|-1 + cot |- - - + --|| | || \2 2 4 / |
|| \ \2 2 4 // | ||--------------------- otherwise |
||---------------------------------------- otherwise | || 2/a b pi\ |
\\ 2 / || 1 + tan |- + - + --| |
\\ \2 2 4 / /
$$\left(- \begin{cases} 1 & \text{for}\: \left(a - b + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\left(- \sin{\left(a - b \right)} + 1\right) \left(\cot^{2}{\left(- \frac{a}{2} + \frac{b}{2} + \frac{\pi}{4} \right)} - 1\right)}{2} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \left(a + b + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{a}{2} + \frac{b}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} + \frac{b}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// / 3*pi\ \ // / 3*pi\ \ || 1 for |a - b + ----| mod 2*pi = 0| || 1 for |a + b + ----| mod 2*pi = 0| || \ 2 / | || \ 2 / | || | || | || 2/b a pi\ | || 2/a b pi\ | - |<-1 + cot |- - - + --| | + |<-1 + tan |- + - + --| | || \2 2 4 / | || \2 2 4 / | ||--------------------- otherwise | ||--------------------- otherwise | || 2/b a pi\ | || 2/a b pi\ | || 1 + cot |- - - + --| | || 1 + tan |- + - + --| | \\ \2 2 4 / / \\ \2 2 4 / /
$$\left(- \begin{cases} 1 & \text{for}\: \left(a - b + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(- \frac{a}{2} + \frac{b}{2} + \frac{\pi}{4} \right)} - 1}{\cot^{2}{\left(- \frac{a}{2} + \frac{b}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \left(a + b + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{a}{2} + \frac{b}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} + \frac{b}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// 0 for (a - b) mod pi = 0\ // 0 for (a + b) mod pi = 0\ || | || | || 2*sin(a - b) | || 2*sin(a + b) | ||------------------------------------ otherwise | ||------------------------------------ otherwise | || / 2 \ | || / 2 \ | - |< | sin (a - b) | | + |< | sin (a + b) | | ||(1 - cos(a - b))*|1 + -------------| | ||(1 - cos(a + b))*|1 + -------------| | || | 4/a - b\| | || | 4/a + b\| | || | 4*sin |-----|| | || | 4*sin |-----|| | || \ \ 2 // | || \ \ 2 // | \\ / \\ /
$$\left(- \begin{cases} 0 & \text{for}\: \left(a - b\right) \bmod \pi = 0 \\\frac{2 \sin{\left(a - b \right)}}{\left(1 + \frac{\sin^{2}{\left(a - b \right)}}{4 \sin^{4}{\left(\frac{a - b}{2} \right)}}\right) \left(- \cos{\left(a - b \right)} + 1\right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(a + b\right) \bmod \pi = 0 \\\frac{2 \sin{\left(a + b \right)}}{\left(1 + \frac{\sin^{2}{\left(a + b \right)}}{4 \sin^{4}{\left(\frac{a + b}{2} \right)}}\right) \left(- \cos{\left(a + b \right)} + 1\right)} & \text{otherwise} \end{cases}\right)$$
// 0 for (a - b) mod pi = 0\ // 0 for (a + b) mod pi = 0\ || | || | || sin(a - b) | || sin(a + b) | ||------------------------------- otherwise | ||------------------------------- otherwise | ||/ 2 \ | ||/ 2 \ | - |<| sin (a - b) | 2/a b\ | + |<| sin (a + b) | 2/a b\ | |||1 + -------------|*sin |- - -| | |||1 + -------------|*sin |- + -| | ||| 4/a b\| \2 2/ | ||| 4/a b\| \2 2/ | ||| 4*sin |- - -|| | ||| 4*sin |- + -|| | ||\ \2 2// | ||\ \2 2// | \\ / \\ /
$$\left(- \begin{cases} 0 & \text{for}\: \left(a - b\right) \bmod \pi = 0 \\\frac{\sin{\left(a - b \right)}}{\left(1 + \frac{\sin^{2}{\left(a - b \right)}}{4 \sin^{4}{\left(\frac{a}{2} - \frac{b}{2} \right)}}\right) \sin^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(a + b\right) \bmod \pi = 0 \\\frac{\sin{\left(a + b \right)}}{\left(1 + \frac{\sin^{2}{\left(a + b \right)}}{4 \sin^{4}{\left(\frac{a}{2} + \frac{b}{2} \right)}}\right) \sin^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// 0 for (a - b) mod pi = 0\ // 0 for (a + b) mod pi = 0\ || | || | ||/ 0 for (a - b) mod pi = 0 | ||/ 0 for (a + b) mod pi = 0 | ||| | ||| | ||| /a b\ | ||| /a b\ | - |<| 2*cot|- - -| | + |<| 2*cot|- + -| | ||< \2 2/ otherwise | ||< \2 2/ otherwise | |||--------------- otherwise | |||--------------- otherwise | ||| 2/a b\ | ||| 2/a b\ | |||1 + cot |- - -| | |||1 + cot |- + -| | \\\ \2 2/ / \\\ \2 2/ /
$$\left(- \begin{cases} 0 & \text{for}\: \left(a - b\right) \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \left(a - b\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} - \frac{b}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(a + b\right) \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \left(a + b\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
/ // a \\ | || 1 for - mod 2*pi = 0|| // b \ // b \ | || 2 || || 0 for - mod pi = 0| || 1 for - mod 2*pi = 0| | || || || 2 | || 2 | | || 2 || || | || | | ||/ 2/a\\ || || /b\ | || 2/b\ | 4*|-1 + 2*|<|-1 + cot |-|| ||*|< 2*cot|-| |*|<-1 + cot |-| | | ||\ \4// || || \4/ | || \4/ | | ||--------------- otherwise || ||----------- otherwise | ||------------ otherwise | | || 2 || || 2/b\ | || 2/b\ | | || / 2/a\\ || ||1 + cot |-| | ||1 + cot |-| | | || |1 + cot |-|| || \\ \4/ / \\ \4/ / \ \\ \ \4// //
$$4 \cdot \left(\left(2 \left(\begin{cases} 1 & \text{for}\: \frac{a}{2} \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) - 1\right) \left(\begin{cases} 0 & \text{for}\: \frac{b}{2} \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{b}{4} \right)}}{\cot^{2}{\left(\frac{b}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \frac{b}{2} \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{b}{4} \right)} - 1}{\cot^{2}{\left(\frac{b}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// 0 for (a - b) mod pi = 0\ // 0 for (a + b) mod pi = 0\
|| | || |
|| /a pi b\ | || /a b pi\ |
|| 2*sec|- - -- - -| | || 2*sec|- + - - --| |
|| \2 2 2/ | || \2 2 2 / |
||--------------------------------- otherwise | ||--------------------------------- otherwise |
- |
$$\left(- \begin{cases} 0 & \text{for}\: \left(a - b\right) \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{a}{2} - \frac{b}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{b}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}\right) \sec{\left(\frac{a}{2} - \frac{b}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(a + b\right) \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{a}{2} + \frac{b}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{a}{2} + \frac{b}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}}\right) \sec{\left(\frac{a}{2} + \frac{b}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// 0 for (a - b) mod pi = 0\ // 0 for (a + b) mod pi = 0\
|| | || |
|| /a b\ | || /a b\ |
|| 2*cos|- - -| | || 2*cos|- + -| |
|| \2 2/ | || \2 2/ |
||-------------------------------------- otherwise | ||-------------------------------------- otherwise |
- |
$$\left(- \begin{cases} 0 & \text{for}\: \left(a - b\right) \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{a}{2} - \frac{b}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{b}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{a}{2} - \frac{b}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(a + b\right) \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} + \frac{b}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{a}{2} + \frac{b}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// 0 for (a - b) mod pi = 0\ // 0 for (a + b) mod pi = 0\
|| | || |
|| /a b\ | || /a b\ |
|| 2*csc|- - -| | || 2*csc|- + -| |
|| \2 2/ | || \2 2/ |
||-------------------------------------- otherwise | ||-------------------------------------- otherwise |
- |
$$\left(- \begin{cases} 0 & \text{for}\: \left(a - b\right) \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{a}{2} - \frac{b}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{b}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{a}{2} + \frac{b}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(a + b\right) \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} - \frac{b}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{a}{2} - \frac{b}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
-Piecewise((0, Mod(a - b = pi, 0)), (2*csc(a/2 - b/2)/((1 + csc(a/2 - b/2)^2/csc(pi/2 + b/2 - a/2)^2)*csc(pi/2 + b/2 - a/2)), True)) + Piecewise((0, Mod(a + b = pi, 0)), (2*csc(a/2 + b/2)/((1 + csc(a/2 + b/2)^2/csc(pi/2 - a/2 - b/2)^2)*csc(pi/2 - a/2 - b/2)), True))