Господин Экзамен
Другие калькуляторы
-
Как пользоваться?
- sin(a+b)-sin(a)*cos(b) если a=-2
- (a^(-5))^4*a^22 если a=3
- -75*b^6+30*b^4-3*b^2 если b=4
- 2*|1-x|-5*|-y| если y=-3
- Разложить многочлен на множители:
- 7*x*y/47+x^5+654+2^4*t^4
- 255+y^2*a^2-25*y^2-9*a^2
- x^2-7*x+10
- x^2+16*x+64
- Общий знаменатель:
- (2*a-1)^2/6*a-6+(a-2)^2/6-6*a
- a+3/a-1-a/1-a
- a+3/a*2+3*a+9-1/a-3+a*3+3*a-9/a*3-27
- (25-3*x)/(x-5)^2-(7*x-x^2)/(5-x)^2
- Умножение столбиком:
- 15*24
- 14*25
- 12*60
- 20*16
- Деление столбиком:
- 26/3
- 30/8
- 11/9
- 18018/6
- Раскрыть скобки в:
- (a+d)*a-d*(a+d)
- (m+4)*(m+4)
- 12*b^3-(4*b^2-1)*(3*b-2)
- (y-2)*(y+3)-(y-1)^2
- Разложение числа на простые множители:
- 2002
- 7140
- 4982
- 4356
- График функции y =:
- -2
- Производная:
- -2
- Интеграл d{x}:
-
-2
-
Идентичные выражения
- sin(a+b)-sin(a)*cos(b) если a=- два
- синус от (a плюс b) минус синус от (a) умножить на косинус от (b) если a равно минус 2
- синус от (a плюс b) минус синус от (a) умножить на косинус от (b) если a равно минус два
- sin(a+b)-sin(a)cos(b) если a=-2
- sina+b-sinacosb если a=-2
- sin(a+b)-sin(a)*cos(b) при a=-2
-
Похожие выражения
- sin(a+b)-sin(a)*cos(b) если a=+2
- sin(a-b)-sin(a)*cos(b) если a=-2
- sin(a+b)+sin(a)*cos(b) если a=-2
sin(a+b)-sin(a)*cos(b) если a=-2
Решение
Вы ввели
[src]
sin(a + b) - sin(a)*cos(b)
$$- \sin{\left(a \right)} \cos{\left(b \right)} + \sin{\left(a + b \right)}$$
sin(a + b) - sin(a)*cos(b)
Подстановка условия
[src]
sin(a + b) - sin(a)*cos(b) при a = -2
подставляем
sin(a + b) - sin(a)*cos(b)
$$- \sin{\left(a \right)} \cos{\left(b \right)} + \sin{\left(a + b \right)}$$
cos(a)*sin(b)
$$\sin{\left(b \right)} \cos{\left(a \right)}$$
переменные
a = -2
$$a = -2$$
cos((-2))*sin(b)
$$\sin{\left(b \right)} \cos{\left((-2) \right)}$$
cos(-2)*sin(b)
$$\sin{\left(b \right)} \cos{\left(-2 \right)}$$
cos(2)*sin(b)
$$\sin{\left(b \right)} \cos{\left(2 \right)}$$
cos(2)*sin(b)
Собрать выражение
[src]
sin(a + b) sin(a - b)
---------- - ----------
2 2
$$- \frac{\sin{\left(a - b \right)}}{2} + \frac{\sin{\left(a + b \right)}}{2}$$
sin(a + b)/2 - sin(a - b)/2
Степени
[src]
/ I*b -I*b\
|e e | / -I*a I*a\
/ I*(-a - b) I*(a + b)\ I*|---- + -----|*\- e + e /
I*\- e + e / \ 2 2 /
- ------------------------------ + ---------------------------------
2 2
$$\frac{i \left(e^{i a} - e^{- i a}\right) \left(\frac{e^{i b}}{2} + \frac{e^{- i b}}{2}\right)}{2} - \frac{i \left(- e^{i \left(- a - b\right)} + e^{i \left(a + b\right)}\right)}{2}$$
-i*(-exp(i*(-a - b)) + exp(i*(a + b)))/2 + i*(exp(i*b)/2 + exp(-i*b)/2)*(-exp(-i*a) + exp(i*a))/2
Тригонометрическая часть
[src]
cos(a)*sin(b)
$$\sin{\left(b \right)} \cos{\left(a \right)}$$
/ pi\
sin(b)*sin|a + --|
\ 2 /
$$\sin{\left(b \right)} \sin{\left(a + \frac{\pi}{2} \right)}$$
/ pi\
cos(a)*cos|b - --|
\ 2 /
$$\cos{\left(a \right)} \cos{\left(b - \frac{\pi}{2} \right)}$$
1
------------------
/ pi\
sec(a)*sec|b - --|
\ 2 /
$$\frac{1}{\sec{\left(a \right)} \sec{\left(b - \frac{\pi}{2} \right)}}$$
1
------------------
/pi \
csc(b)*csc|-- - a|
\2 /
$$\frac{1}{\csc{\left(b \right)} \csc{\left(- a + \frac{\pi}{2} \right)}}$$
sin(a + b) sin(a - b)
---------- - ----------
2 2
$$- \frac{\sin{\left(a - b \right)}}{2} + \frac{\sin{\left(a + b \right)}}{2}$$
/ pi\
- sin(a)*sin|b + --| + sin(a + b)
\ 2 /
$$- \sin{\left(a \right)} \sin{\left(b + \frac{\pi}{2} \right)} + \sin{\left(a + b \right)}$$
1 1 ---------- - ------------- csc(a + b) csc(a)*sec(b)
$$\frac{1}{\csc{\left(a + b \right)}} - \frac{1}{\csc{\left(a \right)} \sec{\left(b \right)}}$$
/ pi\ / pi\
- cos(b)*cos|a - --| + cos|a + b - --|
\ 2 / \ 2 /
$$- \cos{\left(b \right)} \cos{\left(a - \frac{\pi}{2} \right)} + \cos{\left(a + b - \frac{\pi}{2} \right)}$$
1 1
---------- - ------------------
csc(a + b) /pi \
csc(a)*csc|-- - b|
\2 /
$$\frac{1}{\csc{\left(a + b \right)}} - \frac{1}{\csc{\left(a \right)} \csc{\left(- b + \frac{\pi}{2} \right)}}$$
1 1 --------------- - ------------------ / pi\ / pi\ sec|a + b - --| sec(b)*sec|a - --| \ 2 / \ 2 /
$$\frac{1}{\sec{\left(a + b - \frac{\pi}{2} \right)}} - \frac{1}{\sec{\left(b \right)} \sec{\left(a - \frac{\pi}{2} \right)}}$$
1 1
--------------- - -----------------------
csc(pi - a - b) /pi \
csc(pi - a)*csc|-- - b|
\2 /
$$\frac{1}{\csc{\left(- a - b + \pi \right)}} - \frac{1}{\csc{\left(- a + \pi \right)} \csc{\left(- b + \frac{\pi}{2} \right)}}$$
/ 2/a\\ /b\ /b\ 2*|-1 + 2*cos |-||*cos|-|*sin|-| \ \2// \2/ \2/
$$2 \cdot \left(2 \cos^{2}{\left(\frac{a}{2} \right)} - 1\right) \sin{\left(\frac{b}{2} \right)} \cos{\left(\frac{b}{2} \right)}$$
1 1 --------------- - ------------------ /pi \ /pi \ sec|-- - a - b| sec(b)*sec|-- - a| \2 / \2 /
$$\frac{1}{\sec{\left(- a - b + \frac{\pi}{2} \right)}} - \frac{1}{\sec{\left(b \right)} \sec{\left(- a + \frac{\pi}{2} \right)}}$$
/a b\ /a b\
cos(a + b)*tan|- + -| - cos(b)*sin(a) + tan|- + -|
\2 2/ \2 2/
$$- \sin{\left(a \right)} \cos{\left(b \right)} + \cos{\left(a + b \right)} \tan{\left(\frac{a}{2} + \frac{b}{2} \right)} + \tan{\left(\frac{a}{2} + \frac{b}{2} \right)}$$
/a b\
2*tan|- + -|
sin(a + b) + sin(a - b) \2 2/
- ----------------------- + ---------------
2 2/a b\
1 + tan |- + -|
\2 2/
$$- \frac{\sin{\left(a - b \right)} + \sin{\left(a + b \right)}}{2} + \frac{2 \tan{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)} + 1}$$
/a b\
2*tan|- + -|
/ 2/b\\ \2 2/
- |-1 + 2*cos |-||*sin(a) + ---------------
\ \2// 2/a b\
1 + tan |- + -|
\2 2/
$$- \left(2 \cos^{2}{\left(\frac{b}{2} \right)} - 1\right) \sin{\left(a \right)} + \frac{2 \tan{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)} + 1}$$
/ 2/a\\ /b\
2*|1 - tan |-||*tan|-|
\ \2// \2/
---------------------------
/ 2/a\\ / 2/b\\
|1 + tan |-||*|1 + tan |-||
\ \2// \ \2//
$$\frac{2 \cdot \left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \tan{\left(\frac{b}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{b}{2} \right)} + 1\right)}$$
/a b\ /a b\
tan|- + -| tan|- - -|
\2 2/ \2 2/
--------------- - ---------------
2/a b\ 2/a b\
1 + tan |- + -| 1 + tan |- - -|
\2 2/ \2 2/
$$\frac{\tan{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)} + 1} - \frac{\tan{\left(\frac{a}{2} - \frac{b}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} + 1}$$
/a + b\
(1 + cos(a + b))*sin|-----|
\ 2 / /a\
--------------------------- - (1 + cos(a))*cos(b)*tan|-|
/a + b\ \2/
cos|-----|
\ 2 /
$$- \left(\cos{\left(a \right)} + 1\right) \cos{\left(b \right)} \tan{\left(\frac{a}{2} \right)} + \frac{\left(\cos{\left(a + b \right)} + 1\right) \sin{\left(\frac{a + b}{2} \right)}}{\cos{\left(\frac{a + b}{2} \right)}}$$
// 0 for b mod pi = 0\ // 1 for a mod 2*pi = 0\ |< |*|< | \\sin(b) otherwise / \\cos(a) otherwise /
$$\left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\sin{\left(b \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right)$$
2/a + b\
4*sin |-----|
sin(a + b) sin(a - b) \ 2 /
- ---------- - ---------- + ------------------------------
2 2 / 4/a + b\\
| 4*sin |-----||
| \ 2 /|
|1 + -------------|*sin(a + b)
| 2 |
\ sin (a + b) /
$$- \frac{\sin{\left(a - b \right)}}{2} - \frac{\sin{\left(a + b \right)}}{2} + \frac{4 \sin^{2}{\left(\frac{a + b}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{a + b}{2} \right)}}{\sin^{2}{\left(a + b \right)}} + 1\right) \sin{\left(a + b \right)}}$$
2/a b\
4*sin |- + -|
sin(a + b) sin(a - b) \2 2/
- ---------- - ---------- + ------------------------------
2 2 / 4/a b\\
| 4*sin |- + -||
| \2 2/|
|1 + -------------|*sin(a + b)
| 2 |
\ sin (a + b) /
$$- \frac{\sin{\left(a - b \right)}}{2} - \frac{\sin{\left(a + b \right)}}{2} + \frac{4 \sin^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\sin^{2}{\left(a + b \right)}} + 1\right) \sin{\left(a + b \right)}}$$
/a b\ / 2/b\\ /a\
2*tan|- + -| 2*|1 - tan |-||*tan|-|
\2 2/ \ \2// \2/
--------------- - ---------------------------
2/a b\ / 2/a\\ / 2/b\\
1 + tan |- + -| |1 + tan |-||*|1 + tan |-||
\2 2/ \ \2// \ \2//
$$- \frac{2 \cdot \left(- \tan^{2}{\left(\frac{b}{2} \right)} + 1\right) \tan{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{b}{2} \right)} + 1\right)} + \frac{2 \tan{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)} + 1}$$
/a b\ /a\ /b pi\
2*tan|- + -| 4*tan|-|*tan|- + --|
\2 2/ \2/ \2 4 /
--------------- - --------------------------------
2/a b\ / 2/a\\ / 2/b pi\\
1 + tan |- + -| |1 + tan |-||*|1 + tan |- + --||
\2 2/ \ \2// \ \2 4 //
$$\frac{2 \tan{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)} + 1} - \frac{4 \tan{\left(\frac{a}{2} \right)} \tan{\left(\frac{b}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)} + 1\right)}$$
/a b\ /a\ /b pi\
2*cot|- + -| 4*cot|-|*tan|- + --|
\2 2/ \2/ \2 4 /
--------------- - --------------------------------
2/a b\ / 2/a\\ / 2/b pi\\
1 + cot |- + -| |1 + cot |-||*|1 + tan |- + --||
\2 2/ \ \2// \ \2 4 //
$$\frac{2 \cot{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)} + 1} - \frac{4 \tan{\left(\frac{b}{2} + \frac{\pi}{4} \right)} \cot{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)}$$
/pi a b\
2*csc|-- - - - -|
1 1 \2 2 2/
- ------------ - ------------ + ---------------------------------
2*csc(a + b) 2*csc(a - b) / 2/pi a b\\
| csc |-- - - - -||
| \2 2 2/| /a b\
|1 + ----------------|*csc|- + -|
| 2/a b\ | \2 2/
| csc |- + -| |
\ \2 2/ /
$$- \frac{1}{2 \csc{\left(a + b \right)}} - \frac{1}{2 \csc{\left(a - b \right)}} + \frac{2 \csc{\left(- \frac{a}{2} - \frac{b}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} - \frac{b}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}}\right) \csc{\left(\frac{a}{2} + \frac{b}{2} \right)}}$$
/ 2/a b pi\\ 2/b\ / 2/a pi\\ / 2/b\\
|1 - cot |- + - + --||*(1 + sin(a + b)) cos |-|*|1 - cot |- + --||*|1 - tan |-||*(1 + sin(a))
\ \2 2 4 // \2/ \ \2 4 // \ \2//
--------------------------------------- - -----------------------------------------------------
2 2
$$- \frac{\left(- \tan^{2}{\left(\frac{b}{2} \right)} + 1\right) \left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(a \right)} + 1\right) \cos^{2}{\left(\frac{b}{2} \right)}}{2} + \frac{\left(- \cot^{2}{\left(\frac{a}{2} + \frac{b}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(a + b \right)} + 1\right)}{2}$$
// 0 for b mod pi = 0\ // 1 for a mod 2*pi = 0\ || | || | || /b\ | || 2/a\ | || 2*cot|-| | ||-1 + cot |-| | |< \2/ |*|< \2/ | ||----------- otherwise | ||------------ otherwise | || 2/b\ | || 2/a\ | ||1 + cot |-| | ||1 + cot |-| | \\ \2/ / \\ \2/ /
$$\left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{b}{2} \right)}}{\cot^{2}{\left(\frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
/ pi\ / pi\ /a b pi\
cos|a + b - --| cos|a - b - --| 2*cos|- + - - --|
\ 2 / \ 2 / \2 2 2 /
- --------------- - --------------- + ---------------------------------
2 2 / 2/a b pi\\
| cos |- + - - --||
| \2 2 2 /| /a b\
|1 + ----------------|*cos|- + -|
| 2/a b\ | \2 2/
| cos |- + -| |
\ \2 2/ /
$$- \frac{\cos{\left(a - b - \frac{\pi}{2} \right)}}{2} - \frac{\cos{\left(a + b - \frac{\pi}{2} \right)}}{2} + \frac{2 \cos{\left(\frac{a}{2} + \frac{b}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{a}{2} + \frac{b}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}}\right) \cos{\left(\frac{a}{2} + \frac{b}{2} \right)}}$$
/ 1 \
2*|1 - -------|
| 2/b\|
| cot |-||
2 \ \2//
---------------------------- - ----------------------------------
/ 1 \ /a b\ / 1 \ / 1 \ /a\
|1 + -----------|*cot|- + -| |1 + -------|*|1 + -------|*cot|-|
| 2/a b\| \2 2/ | 2/a\| | 2/b\| \2/
| cot |- + -|| | cot |-|| | cot |-||
\ \2 2// \ \2// \ \2//
$$\frac{2}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}}\right) \cot{\left(\frac{a}{2} + \frac{b}{2} \right)}} - \frac{2 \cdot \left(1 - \frac{1}{\cot^{2}{\left(\frac{b}{2} \right)}}\right)}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \left(1 + \frac{1}{\cot^{2}{\left(\frac{b}{2} \right)}}\right) \cot{\left(\frac{a}{2} \right)}}$$
/a b\
2*sec|- + -|
1 1 \2 2/
- ----------------- - ----------------- + --------------------------------------
/ pi\ / pi\ / 2/a b\ \
2*sec|a + b - --| 2*sec|a - b - --| | sec |- + -| |
\ 2 / \ 2 / | \2 2/ | /a b pi\
|1 + ----------------|*sec|- + - - --|
| 2/a b pi\| \2 2 2 /
| sec |- + - - --||
\ \2 2 2 //
$$- \frac{1}{2 \sec{\left(a + b - \frac{\pi}{2} \right)}} - \frac{1}{2 \sec{\left(a - b - \frac{\pi}{2} \right)}} + \frac{2 \sec{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} + \frac{b}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{a}{2} + \frac{b}{2} - \frac{\pi}{2} \right)}}$$
// 0 for a mod pi = 0\ // 1 for b mod 2*pi = 0\ // 0 for (a + b) mod pi = 0\ - |< |*|< | + |< | \\sin(a) otherwise / \\cos(b) otherwise / \\sin(a + b) otherwise /
$$\left(- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\cos{\left(b \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \left(a + b\right) \bmod \pi = 0 \\\sin{\left(a + b \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for b mod 2*pi = 0\
// 0 for a mod pi = 0\ || | // 0 for (a + b) mod pi = 0\
- |< |*|< / pi\ | + |< |
\\sin(a) otherwise / ||sin|b + --| otherwise | \\sin(a + b) otherwise /
\\ \ 2 / /
$$\left(- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\sin{\left(b + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \left(a + b\right) \bmod \pi = 0 \\\sin{\left(a + b \right)} & \text{otherwise} \end{cases}\right)$$
2/a b pi\ / 2/b\\ / 2/a pi\\
-1 + tan |- + - + --| |-1 + cot |-||*|-1 + tan |- + --||
\2 2 4 / \ \2// \ \2 4 //
--------------------- - ----------------------------------
2/a b pi\ / 2/b\\ / 2/a pi\\
1 + tan |- + - + --| |1 + cot |-||*|1 + tan |- + --||
\2 2 4 / \ \2// \ \2 4 //
$$- \frac{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1\right) \left(\cot^{2}{\left(\frac{b}{2} \right)} - 1\right)}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(\frac{b}{2} \right)} + 1\right)} + \frac{\tan^{2}{\left(\frac{a}{2} + \frac{b}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} + \frac{b}{2} + \frac{\pi}{4} \right)} + 1}$$
/ 0 for (a + b) mod pi = 0 / 0 for (a - b) mod pi = 0 /a b\
< < 2*tan|- + -|
\sin(a + b) otherwise \sin(a - b) otherwise \2 2/
- ----------------------------------- - ----------------------------------- + ---------------
2 2 2/a b\
1 + tan |- + -|
\2 2/
$$\left(- \frac{\begin{cases} 0 & \text{for}\: \left(a - b\right) \bmod \pi = 0 \\\sin{\left(a - b \right)} & \text{otherwise} \end{cases}}{2}\right) - \left(\frac{\begin{cases} 0 & \text{for}\: \left(a + b\right) \bmod \pi = 0 \\\sin{\left(a + b \right)} & \text{otherwise} \end{cases}}{2}\right) + \frac{2 \tan{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)} + 1}$$
2/a b pi\ / 2/a pi\\ / 2/b\\
1 - cot |- + - + --| |1 - cot |- + --||*|1 - tan |-||
\2 2 4 / \ \2 4 // \ \2//
-------------------- - --------------------------------
2/a b pi\ / 2/a pi\\ / 2/b\\
1 + cot |- + - + --| |1 + cot |- + --||*|1 + tan |-||
\2 2 4 / \ \2 4 // \ \2//
$$- \frac{\left(- \tan^{2}{\left(\frac{b}{2} \right)} + 1\right) \left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)}{\left(\tan^{2}{\left(\frac{b}{2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)} + \frac{- \cot^{2}{\left(\frac{a}{2} + \frac{b}{2} + \frac{\pi}{4} \right)} + 1}{\cot^{2}{\left(\frac{a}{2} + \frac{b}{2} + \frac{\pi}{4} \right)} + 1}$$
// 0 for a mod pi = 0\ // 0 for (a + b) mod pi = 0\ || | // 1 for b mod 2*pi = 0\ || | - |< / pi\ |*|< | + |< / pi\ | ||cos|a - --| otherwise | \\cos(b) otherwise / ||cos|a + b - --| otherwise | \\ \ 2 / / \\ \ 2 / /
$$\left(- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\cos{\left(b \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \left(a + b\right) \bmod \pi = 0 \\\cos{\left(a + b - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for b mod 2*pi = 0\
// 0 for a mod pi = 0\ || | // 0 for (a + b) mod pi = 0\
|| | || 1 | || |
- |< 1 |*|<----------- otherwise | + |< 1 |
||------ otherwise | || /pi \ | ||---------- otherwise |
\\csc(a) / ||csc|-- - b| | \\csc(a + b) /
\\ \2 / /
$$\left(- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\csc{\left(a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- b + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \left(a + b\right) \bmod \pi = 0 \\\frac{1}{\csc{\left(a + b \right)}} & \text{otherwise} \end{cases}\right)$$
// 0 for (a + b) mod pi = 0\
|| |
// 0 for a mod pi = 0\ // 1 for b mod 2*pi = 0\ ||1 - cos(a + b) |
- |< |*|< | + |<-------------- otherwise |
\\sin(a) otherwise / \\cos(b) otherwise / || /a b\ |
|| tan|- + -| |
\\ \2 2/ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\cos{\left(b \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \left(a + b\right) \bmod \pi = 0 \\\frac{- \cos{\left(a + b \right)} + 1}{\tan{\left(\frac{a}{2} + \frac{b}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\ // 0 for (a + b) mod pi = 0\ || | // 1 for b mod 2*pi = 0\ || | || 1 | || | || 1 | - |<----------- otherwise |*|< 1 | + |<--------------- otherwise | || / pi\ | ||------ otherwise | || / pi\ | ||sec|a - --| | \\sec(b) / ||sec|a + b - --| | \\ \ 2 / / \\ \ 2 / /
$$\left(- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(b \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \left(a + b\right) \bmod \pi = 0 \\\frac{1}{\sec{\left(a + b - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// 0 for (a + b) mod pi = 0\
|| |
|| /a b\ |
// 0 for a mod pi = 0\ // 1 for b mod 2*pi = 0\ || 2*tan|- + -| |
- |< |*|< | + |< \2 2/ |
\\sin(a) otherwise / \\cos(b) otherwise / ||--------------- otherwise |
|| 2/a b\ |
||1 + tan |- + -| |
\\ \2 2/ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\cos{\left(b \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \left(a + b\right) \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// / pi\ \
|| 0 for |b + --| mod pi = 0|
// 0 for a mod pi = 0\ || \ 2 / | // 0 for (a + b) mod pi = 0\
- |< |*|< | + |< |
\\sin(a) otherwise / || /b pi\ | \\sin(a + b) otherwise /
||(1 + sin(b))*cot|- + --| otherwise |
\\ \2 4 / /
$$\left(- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(b + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(b \right)} + 1\right) \cot{\left(\frac{b}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \left(a + b\right) \bmod \pi = 0 \\\sin{\left(a + b \right)} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\ // 0 for (a + b) mod pi = 0\ || | || | ||1 - cos(a) | // 1 for b mod 2*pi = 0\ ||1 - cos(a + b) | - |<---------- otherwise |*|< | + |<-------------- otherwise | || /a\ | \\cos(b) otherwise / || /a b\ | || tan|-| | || tan|- + -| | \\ \2/ / \\ \2 2/ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{- \cos{\left(a \right)} + 1}{\tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\cos{\left(b \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \left(a + b\right) \bmod \pi = 0 \\\frac{- \cos{\left(a + b \right)} + 1}{\tan{\left(\frac{a}{2} + \frac{b}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// 0 for (a + b) mod pi = 0\
|| |
|| 2 |
// 0 for a mod pi = 0\ // 1 for b mod 2*pi = 0\ ||---------------------------- otherwise |
- |< |*|< | + |
$$\left(- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\cos{\left(b \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \left(a + b\right) \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}}\right) \tan{\left(\frac{a}{2} + \frac{b}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
/ 4/b\\
| 4*sin |-||
2/a\ | \2/|
2/a + b\ 4*sin |-|*|1 - ---------|
4*sin |-----| \2/ | 2 |
\ 2 / \ sin (b) /
------------------------------ - --------------------------------------
/ 4/a + b\\ / 4/a\\ / 4/b\\
| 4*sin |-----|| | 4*sin |-|| | 4*sin |-||
| \ 2 /| | \2/| | \2/|
|1 + -------------|*sin(a + b) |1 + ---------|*|1 + ---------|*sin(a)
| 2 | | 2 | | 2 |
\ sin (a + b) / \ sin (a) / \ sin (b) /
$$\frac{4 \sin^{2}{\left(\frac{a + b}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{a + b}{2} \right)}}{\sin^{2}{\left(a + b \right)}} + 1\right) \sin{\left(a + b \right)}} - \frac{4 \left(- \frac{4 \sin^{4}{\left(\frac{b}{2} \right)}}{\sin^{2}{\left(b \right)}} + 1\right) \sin^{2}{\left(\frac{a}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right) \left(\frac{4 \sin^{4}{\left(\frac{b}{2} \right)}}{\sin^{2}{\left(b \right)}} + 1\right) \sin{\left(a \right)}}$$
/ 4/b\\
| 4*sin |-||
2/a\ | \2/|
2/a b\ 4*sin |-|*|1 - ---------|
4*sin |- + -| \2/ | 2 |
\2 2/ \ sin (b) /
------------------------------ - --------------------------------------
/ 4/a b\\ / 4/a\\ / 4/b\\
| 4*sin |- + -|| | 4*sin |-|| | 4*sin |-||
| \2 2/| | \2/| | \2/|
|1 + -------------|*sin(a + b) |1 + ---------|*|1 + ---------|*sin(a)
| 2 | | 2 | | 2 |
\ sin (a + b) / \ sin (a) / \ sin (b) /
$$\frac{4 \sin^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\sin^{2}{\left(a + b \right)}} + 1\right) \sin{\left(a + b \right)}} - \frac{4 \left(- \frac{4 \sin^{4}{\left(\frac{b}{2} \right)}}{\sin^{2}{\left(b \right)}} + 1\right) \sin^{2}{\left(\frac{a}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right) \left(\frac{4 \sin^{4}{\left(\frac{b}{2} \right)}}{\sin^{2}{\left(b \right)}} + 1\right) \sin{\left(a \right)}}$$
// / 3*pi\ \
|| 1 for |a + b + ----| mod 2*pi = 0|
|| \ 2 / |
// / 3*pi\ \ || |
// 1 for b mod 2*pi = 0\ || 1 for |a + ----| mod 2*pi = 0| || 2/a b pi\ |
- |< |*|< \ 2 / | + |<-1 + tan |- + - + --| |
\\cos(b) otherwise / || | || \2 2 4 / |
\\sin(a) otherwise / ||--------------------- otherwise |
|| 2/a b pi\ |
|| 1 + tan |- + - + --| |
\\ \2 2 4 / /
$$\left(- \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\cos{\left(b \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \left(a + b + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{a}{2} + \frac{b}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} + \frac{b}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\ // 1 for b mod 2*pi = 0\ // 0 for (a + b) mod pi = 0\ || | || | || | || /a\ | || 2/b\ | || /a b\ | || 2*cot|-| | ||-1 + cot |-| | || 2*cot|- + -| | - |< \2/ |*|< \2/ | + |< \2 2/ | ||----------- otherwise | ||------------ otherwise | ||--------------- otherwise | || 2/a\ | || 2/b\ | || 2/a b\ | ||1 + cot |-| | ||1 + cot |-| | ||1 + cot |- + -| | \\ \2/ / \\ \2/ / \\ \2 2/ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{b}{2} \right)} - 1}{\cot^{2}{\left(\frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \left(a + b\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\ // 1 for b mod 2*pi = 0\ // 0 for (a + b) mod pi = 0\ || | || | || | || /a\ | || 2/b\ | || /a b\ | || 2*tan|-| | ||1 - tan |-| | || 2*tan|- + -| | - |< \2/ |*|< \2/ | + |< \2 2/ | ||----------- otherwise | ||----------- otherwise | ||--------------- otherwise | || 2/a\ | || 2/b\ | || 2/a b\ | ||1 + tan |-| | ||1 + tan |-| | ||1 + tan |- + -| | \\ \2/ / \\ \2/ / \\ \2 2/ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{b}{2} \right)} + 1}{\tan^{2}{\left(\frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \left(a + b\right) \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
/ 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/ 2
- ---------------------------------------- - ---------------------------------------- + ----------------------------
2 2 / 1 \ /a b\
|1 + -----------|*cot|- + -|
| 2/a b\| \2 2/
| cot |- + -||
\ \2 2//
$$\left(- \frac{\begin{cases} 0 & \text{for}\: \left(a - b\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} - \frac{b}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases}}{2}\right) - \left(\frac{\begin{cases} 0 & \text{for}\: \left(a + b\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases}}{2}\right) + \frac{2}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}}\right) \cot{\left(\frac{a}{2} + \frac{b}{2} \right)}}$$
// 0 for a mod pi = 0\ // 1 for b mod 2*pi = 0\ // 0 for (a + b) mod pi = 0\
|| | || | || |
- |
$$\left(- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\cos{\left(b \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \left(a + b\right) \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \left(a + b\right) \bmod \pi = 0 \\\sin{\left(a + b \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
// / pi\ \
// 0 for a mod pi = 0\ || 0 for |b + --| mod pi = 0| // 0 for (a + b) mod pi = 0\
|| | || \ 2 / | || |
|| /a\ | || | || /a b\ |
|| 2*cot|-| | || /b pi\ | || 2*cot|- + -| |
- |< \2/ |*|< 2*cot|- + --| | + |< \2 2/ |
||----------- otherwise | || \2 4 / | ||--------------- otherwise |
|| 2/a\ | ||---------------- otherwise | || 2/a b\ |
||1 + cot |-| | || 2/b pi\ | ||1 + cot |- + -| |
\\ \2/ / ||1 + cot |- + --| | \\ \2 2/ /
\\ \2 4 / /
$$\left(- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(b + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{b}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \left(a + b\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// 1 for b mod 2*pi = 0\
|| |
// 0 for a mod pi = 0\ || 1 | // 0 for (a + b) mod pi = 0\
|| | ||-1 + ------- | || |
|| 2 | || 2/b\ | || 2 |
||-------------------- otherwise | || tan |-| | ||---------------------------- otherwise |
- |
$$\left(- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right) \tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{b}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{b}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \left(a + b\right) \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}}\right) \tan{\left(\frac{a}{2} + \frac{b}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
/ 2/b\ \
| sec |-| |
| \2/ | /a\
2*|1 - ------------|*sec|-|
/a b\ | 2/b pi\| \2/
2*sec|- + -| | sec |- - --||
\2 2/ \ \2 2 //
-------------------------------------- - -------------------------------------------------
/ 2/a b\ \ / 2/a\ \ / 2/b\ \
| sec |- + -| | | sec |-| | | sec |-| |
| \2 2/ | /a b pi\ | \2/ | | \2/ | /a pi\
|1 + ----------------|*sec|- + - - --| |1 + ------------|*|1 + ------------|*sec|- - --|
| 2/a b pi\| \2 2 2 / | 2/a pi\| | 2/b pi\| \2 2 /
| sec |- + - - --|| | sec |- - --|| | sec |- - --||
\ \2 2 2 // \ \2 2 // \ \2 2 //
$$\frac{2 \sec{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} + \frac{b}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{a}{2} + \frac{b}{2} - \frac{\pi}{2} \right)}} - \frac{2 \left(- \frac{\sec^{2}{\left(\frac{b}{2} \right)}}{\sec^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{a}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \left(\frac{\sec^{2}{\left(\frac{b}{2} \right)}}{\sec^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}$$
/ 2/b pi\\
| cos |- - --||
| \2 2 /| /a pi\
2*|1 - ------------|*cos|- - --|
/a b pi\ | 2/b\ | \2 2 /
2*cos|- + - - --| | cos |-| |
\2 2 2 / \ \2/ /
--------------------------------- - --------------------------------------------
/ 2/a b pi\\ / 2/a pi\\ / 2/b pi\\
| cos |- + - - --|| | cos |- - --|| | cos |- - --||
| \2 2 2 /| /a b\ | \2 2 /| | \2 2 /| /a\
|1 + ----------------|*cos|- + -| |1 + ------------|*|1 + ------------|*cos|-|
| 2/a b\ | \2 2/ | 2/a\ | | 2/b\ | \2/
| cos |- + -| | | cos |-| | | cos |-| |
\ \2 2/ / \ \2/ / \ \2/ /
$$\frac{2 \cos{\left(\frac{a}{2} + \frac{b}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{a}{2} + \frac{b}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}}\right) \cos{\left(\frac{a}{2} + \frac{b}{2} \right)}} - \frac{2 \cdot \left(1 - \frac{\cos^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{b}{2} \right)}}\right) \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right) \left(1 + \frac{\cos^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{b}{2} \right)}}\right) \cos{\left(\frac{a}{2} \right)}}$$
/ 2/pi b\\
| csc |-- - -||
| \2 2/| /pi a\
2*|1 - ------------|*csc|-- - -|
/pi a b\ | 2/b\ | \2 2/
2*csc|-- - - - -| | csc |-| |
\2 2 2/ \ \2/ /
--------------------------------- - --------------------------------------------
/ 2/pi a b\\ / 2/pi a\\ / 2/pi b\\
| csc |-- - - - -|| | csc |-- - -|| | csc |-- - -||
| \2 2 2/| /a b\ | \2 2/| | \2 2/| /a\
|1 + ----------------|*csc|- + -| |1 + ------------|*|1 + ------------|*csc|-|
| 2/a b\ | \2 2/ | 2/a\ | | 2/b\ | \2/
| csc |- + -| | | csc |-| | | csc |-| |
\ \2 2/ / \ \2/ / \ \2/ /
$$\frac{2 \csc{\left(- \frac{a}{2} - \frac{b}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} - \frac{b}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}}\right) \csc{\left(\frac{a}{2} + \frac{b}{2} \right)}} - \frac{2 \cdot \left(1 - \frac{\csc^{2}{\left(- \frac{b}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{b}{2} \right)}}\right) \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right) \left(1 + \frac{\csc^{2}{\left(- \frac{b}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{b}{2} \right)}}\right) \csc{\left(\frac{a}{2} \right)}}$$
// / 3*pi\ \ // / 3*pi\ \
// 1 for b mod 2*pi = 0\ || 1 for |a + ----| mod 2*pi = 0| || 1 for |a + b + ----| mod 2*pi = 0|
|| | || \ 2 / | || \ 2 / |
|| 2/b\ | || | || |
||-1 + cot |-| | || 2/a pi\ | || 2/a b pi\ |
- |< \2/ |*|<-1 + tan |- + --| | + |<-1 + tan |- + - + --| |
||------------ otherwise | || \2 4 / | || \2 2 4 / |
|| 2/b\ | ||----------------- otherwise | ||--------------------- otherwise |
||1 + cot |-| | || 2/a pi\ | || 2/a b pi\ |
\\ \2/ / || 1 + tan |- + --| | || 1 + tan |- + - + --| |
\\ \2 4 / / \\ \2 2 4 / /
$$\left(- \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{b}{2} \right)} - 1}{\cot^{2}{\left(\frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \left(a + b + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{a}{2} + \frac{b}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} + \frac{b}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\ // 0 for (a + b) mod pi = 0\ || | || | || 2*sin(a) | // 1 for b mod 2*pi = 0\ || 2*sin(a + b) | ||---------------------------- otherwise | || | ||------------------------------------ otherwise | || / 2 \ | || 2 | || / 2 \ | - |< | sin (a) | |*|< -4 + 4*sin (b) + 4*cos(b) | + |< | sin (a + b) | | ||(1 - cos(a))*|1 + ---------| | ||--------------------------- otherwise | ||(1 - cos(a + b))*|1 + -------------| | || | 4/a\| | || 2 2 | || | 4/a + b\| | || | 4*sin |-|| | \\2*(1 - cos(b)) + 2*sin (b) / || | 4*sin |-----|| | || \ \2// | || \ \ 2 // | \\ / \\ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \sin{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \left(- \cos{\left(a \right)} + 1\right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{4 \sin^{2}{\left(b \right)} + 4 \cos{\left(b \right)} - 4}{2 \left(- \cos{\left(b \right)} + 1\right)^{2} + 2 \sin^{2}{\left(b \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \left(a + b\right) \bmod \pi = 0 \\\frac{2 \sin{\left(a + b \right)}}{\left(1 + \frac{\sin^{2}{\left(a + b \right)}}{4 \sin^{4}{\left(\frac{a + b}{2} \right)}}\right) \left(- \cos{\left(a + b \right)} + 1\right)} & \text{otherwise} \end{cases}\right)$$
// 1 for b mod 2*pi = 0\
|| |
// 0 for a mod pi = 0\ || 2 | // 0 for (a + b) mod pi = 0\
|| | || sin (b) | || |
|| sin(a) | ||-1 + --------- | || sin(a + b) |
||----------------------- otherwise | || 4/b\ | ||------------------------------- otherwise |
||/ 2 \ | || 4*sin |-| | ||/ 2 \ |
- |<| sin (a) | 2/a\ |*|< \2/ | + |<| sin (a + b) | 2/a b\ |
|||1 + ---------|*sin |-| | ||-------------- otherwise | |||1 + -------------|*sin |- + -| |
||| 4/a\| \2/ | || 2 | ||| 4/a b\| \2 2/ |
||| 4*sin |-|| | || sin (b) | ||| 4*sin |- + -|| |
||\ \2// | ||1 + --------- | ||\ \2 2// |
\\ / || 4/b\ | \\ /
|| 4*sin |-| |
\\ \2/ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{\sin{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \sin^{2}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(b \right)}}{4 \sin^{4}{\left(\frac{b}{2} \right)}}}{1 + \frac{\sin^{2}{\left(b \right)}}{4 \sin^{4}{\left(\frac{b}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \left(a + b\right) \bmod \pi = 0 \\\frac{\sin{\left(a + b \right)}}{\left(1 + \frac{\sin^{2}{\left(a + b \right)}}{4 \sin^{4}{\left(\frac{a}{2} + \frac{b}{2} \right)}}\right) \sin^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\ // 1 for b mod 2*pi = 0\ // 0 for (a + b) mod pi = 0\ || | || | || | ||/ 0 for a mod pi = 0 | ||/ 1 for b mod 2*pi = 0 | ||/ 0 for (a + b) mod pi = 0 | ||| | ||| | ||| | ||| /a\ | ||| 2/b\ | ||| /a b\ | - |<| 2*cot|-| |*|<|-1 + cot |-| | + |<| 2*cot|- + -| | ||< \2/ otherwise | ||< \2/ otherwise | ||< \2 2/ otherwise | |||----------- otherwise | |||------------ otherwise | |||--------------- otherwise | ||| 2/a\ | ||| 2/b\ | ||| 2/a b\ | |||1 + cot |-| | |||1 + cot |-| | |||1 + cot |- + -| | \\\ \2/ / \\\ \2/ / \\\ \2 2/ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{b}{2} \right)} - 1}{\cot^{2}{\left(\frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \left(a + b\right) \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \left(a + b\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
// 1 for b mod 2*pi = 0\
|| |
// 0 for a mod pi = 0\ || 2/b\ | // 0 for (a + b) mod pi = 0\
|| | || cos |-| | || |
|| /a\ | || \2/ | || /a b\ |
|| 2*cos|-| | ||-1 + ------------ | || 2*cos|- + -| |
|| \2/ | || 2/b pi\ | || \2 2/ |
||------------------------------ otherwise | || cos |- - --| | ||-------------------------------------- otherwise |
- |
$$\left(- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{a}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{b}{2} \right)}}{\cos^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{b}{2} \right)}}{\cos^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \left(a + b\right) \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} + \frac{b}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{a}{2} + \frac{b}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// 1 for b mod 2*pi = 0\
|| |
// 0 for a mod pi = 0\ || 2/b pi\ | // 0 for (a + b) mod pi = 0\
|| | || sec |- - --| | || |
|| /a pi\ | || \2 2 / | || /a b pi\ |
|| 2*sec|- - --| | ||-1 + ------------ | || 2*sec|- + - - --| |
|| \2 2 / | || 2/b\ | || \2 2 2 / |
||------------------------- otherwise | || sec |-| | ||--------------------------------- otherwise |
- |
$$\left(- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right) \sec{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{b}{2} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{b}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \left(a + b\right) \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{a}{2} + \frac{b}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{a}{2} + \frac{b}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}}\right) \sec{\left(\frac{a}{2} + \frac{b}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// 1 for b mod 2*pi = 0\
|| |
// 0 for a mod pi = 0\ || 2/b\ | // 0 for (a + b) mod pi = 0\
|| | || csc |-| | || |
|| /a\ | || \2/ | || /a b\ |
|| 2*csc|-| | ||-1 + ------------ | || 2*csc|- + -| |
|| \2/ | || 2/pi b\ | || \2 2/ |
||------------------------------ otherwise | || csc |-- - -| | ||-------------------------------------- otherwise |
- |
$$\left(- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{a}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{b}{2} \right)}}{\csc^{2}{\left(- \frac{b}{2} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{b}{2} \right)}}{\csc^{2}{\left(- \frac{b}{2} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \left(a + b\right) \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} - \frac{b}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{a}{2} - \frac{b}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
-Piecewise((0, Mod(a = pi, 0)), (2*csc(a/2)/((1 + csc(a/2)^2/csc(pi/2 - a/2)^2)*csc(pi/2 - a/2)), True))*Piecewise((1, Mod(b = 2*pi, 0)), ((-1 + csc(b/2)^2/csc(pi/2 - b/2)^2)/(1 + csc(b/2)^2/csc(pi/2 - b/2)^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))