Господин Экзамен
Другие калькуляторы
-
Как пользоваться?
- 7*b-3*x+b+2*x если x=1/2
- 2*x^3+9*x^2+11*x+6 если x=-3
- cos(a-b)-cos(a)*cos(b) если a=1/2
- 12*m-5*(2*n-n+1-(-1)*(-5*n)) если m=4
- Разложить многочлен на множители:
- 121*x^2-66*x+9
- 49-9*n^2+6*m*n-m^2
- 8-72*x^6*y^2
- 27-x^3
- Общий знаменатель:
- n^2/n-4-8*n-16/n-4
- t/(p-5)^2-10/(5-p)^2
- (sqrt(m)/n-sqrt(m*n)-sqrt(n)/sqrt(m*n)-m)*sqrt(m*n)/sqrt(n)+sqrt(m)
- 3*x/y+1*y+y2/9
- Умножение столбиком:
- 66*66
- 85*20
- 44*42
- 38*756
- Деление столбиком:
- 942/9
- 329/5
- 74185/5
- 614/6
- Раскрыть скобки в:
- 7*(p-3*q)-8*(3*p+q)
- x*y^2+x^2*y
- n*(m-n)+2*m*(n-m)
- (12-3)*(12+3)
- Разложение числа на простые множители:
- 10350
- 3540
- 44100
- 57
- График функции y =:
- 1/2
- Производная:
- 1/2
- Интеграл d{x}:
-
1/2
-
Идентичные выражения
- cos(a-b)-cos(a)*cos(b) если a= один / два
- косинус от (a минус b) минус косинус от (a) умножить на косинус от (b) если a равно 1 делить на 2
- косинус от (a минус b) минус косинус от (a) умножить на косинус от (b) если a равно один делить на два
- cos(a-b)-cos(a)cos(b) если a=1/2
- cosa-b-cosacosb если a=1/2
- cos(a-b)-cos(a)*cos(b) при a=1/2
- cos(a-b)-cos(a)*cos(b) если a=1 разделить на 2
-
Похожие выражения
- cos(a-b)-cos(acos(b)) если a=1/2
- cos(a+b)-cos(a)*cos(b) если a=1/2
- cos(a-b)+cos(a)*cos(b) если a=1/2
cos(a-b)-cos(a)*cos(b) если a=1/2
Решение
Вы ввели
[src]
cos(a - b) - cos(a)*cos(b)
$$- \cos{\left(a \right)} \cos{\left(b \right)} + \cos{\left(a - b \right)}$$
cos(a - b) - cos(a)*cos(b)
Подстановка условия
[src]
cos(a - b) - cos(a)*cos(b) при a = 1/2
подставляем
cos(a - b) - cos(a)*cos(b)
$$- \cos{\left(a \right)} \cos{\left(b \right)} + \cos{\left(a - b \right)}$$
sin(a)*sin(b)
$$\sin{\left(a \right)} \sin{\left(b \right)}$$
переменные
a = 1/2
$$a = \frac{1}{2}$$
sin((1/2))*sin(b)
$$\sin{\left((1/2) \right)} \sin{\left(b \right)}$$
sin(1/2)*sin(b)
$$\sin{\left(\frac{1}{2} \right)} \sin{\left(b \right)}$$
sin(1/2)*sin(b)
Собрать выражение
[src]
cos(a - b) cos(a + b)
---------- - ----------
2 2
$$\frac{\cos{\left(a - b \right)}}{2} - \frac{\cos{\left(a + b \right)}}{2}$$
cos(a - b)/2 - cos(a + b)/2
Степени
[src]
I*(a - b) I*(b - a) / I*a -I*a\ / I*b -I*b\
e e |e e | |e e |
---------- + ---------- - |---- + -----|*|---- + -----|
2 2 \ 2 2 / \ 2 2 /
$$- \left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2}\right) \left(\frac{e^{i b}}{2} + \frac{e^{- i b}}{2}\right) + \frac{e^{i \left(- a + b\right)}}{2} + \frac{e^{i \left(a - b\right)}}{2}$$
exp(i*(a - b))/2 + exp(i*(b - a))/2 - (exp(i*a)/2 + exp(-i*a)/2)*(exp(i*b)/2 + exp(-i*b)/2)
Тригонометрическая часть
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sin(a)*sin(b)
$$\sin{\left(a \right)} \sin{\left(b \right)}$$
1 ------------- csc(a)*csc(b)
$$\frac{1}{\csc{\left(a \right)} \csc{\left(b \right)}}$$
cos(a - b) cos(a + b)
---------- - ----------
2 2
$$\frac{\cos{\left(a - b \right)}}{2} - \frac{\cos{\left(a + b \right)}}{2}$$
/ pi\ / pi\ cos|a - --|*cos|b - --| \ 2 / \ 2 /
$$\cos{\left(a - \frac{\pi}{2} \right)} \cos{\left(b - \frac{\pi}{2} \right)}$$
1 ----------------------- / pi\ / pi\ sec|a - --|*sec|b - --| \ 2 / \ 2 /
$$\frac{1}{\sec{\left(a - \frac{\pi}{2} \right)} \sec{\left(b - \frac{\pi}{2} \right)}}$$
1 1 ---------- - ------------- sec(a - b) sec(a)*sec(b)
$$\frac{1}{\sec{\left(a - b \right)}} - \frac{1}{\sec{\left(a \right)} \sec{\left(b \right)}}$$
/ pi\ / pi\ / pi \
- sin|a + --|*sin|b + --| + sin|a + -- - b|
\ 2 / \ 2 / \ 2 /
$$- \sin{\left(a + \frac{\pi}{2} \right)} \sin{\left(b + \frac{\pi}{2} \right)} + \sin{\left(a - b + \frac{\pi}{2} \right)}$$
/a\ /b\ /a\ /b\
4*cos|-|*cos|-|*sin|-|*sin|-|
\2/ \2/ \2/ \2/
$$4 \sin{\left(\frac{a}{2} \right)} \sin{\left(\frac{b}{2} \right)} \cos{\left(\frac{a}{2} \right)} \cos{\left(\frac{b}{2} \right)}$$
1 1 --------------- - ----------------------- / pi \ /pi \ /pi \ csc|b + -- - a| csc|-- - a|*csc|-- - b| \ 2 / \2 / \2 /
$$\frac{1}{\csc{\left(- a + b + \frac{\pi}{2} \right)}} - \frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)} \csc{\left(- b + \frac{\pi}{2} \right)}}$$
/a\ /b\
4*tan|-|*tan|-|
\2/ \2/
---------------------------
/ 2/a\\ / 2/b\\
|1 + tan |-||*|1 + tan |-||
\ \2// \ \2//
$$\frac{4 \tan{\left(\frac{a}{2} \right)} \tan{\left(\frac{b}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{b}{2} \right)} + 1\right)}$$
2/a b\
1 - tan |- - -|
cos(a + b) + cos(a - b) \2 2/
- ----------------------- + ---------------
2 2/a b\
1 + tan |- - -|
\2 2/
$$- \frac{\cos{\left(a - b \right)} + \cos{\left(a + b \right)}}{2} + \frac{- \tan^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} + 1}{\tan^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} + 1}$$
// 0 for a mod pi = 0\ // 0 for b mod pi = 0\ |< |*|< | \\sin(a) otherwise / \\sin(b) otherwise /
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\sin{\left(b \right)} & \text{otherwise} \end{cases}\right)$$
1 cos(a + b) cos(a - b) sin(a - b) --------------- - ---------- - ---------- - --------------- /b a pi\ 2 2 /b a pi\ tan|- - - + --| tan|- - - + --| \2 2 4 / \2 2 4 /
$$- \frac{\cos{\left(a - b \right)}}{2} - \frac{\cos{\left(a + b \right)}}{2} - \frac{\sin{\left(a - b \right)}}{\tan{\left(- \frac{a}{2} + \frac{b}{2} + \frac{\pi}{4} \right)}} + \frac{1}{\tan{\left(- \frac{a}{2} + \frac{b}{2} + \frac{\pi}{4} \right)}}$$
4/a - b\
4*sin |-----|
\ 2 /
1 - -------------
2
cos(a + b) cos(a - b) sin (a - b)
- ---------- - ---------- + -----------------
2 2 4/a - b\
4*sin |-----|
\ 2 /
1 + -------------
2
sin (a - b)
$$- \frac{\cos{\left(a - b \right)}}{2} - \frac{\cos{\left(a + b \right)}}{2} + \frac{- \frac{4 \sin^{4}{\left(\frac{a - b}{2} \right)}}{\sin^{2}{\left(a - b \right)}} + 1}{\frac{4 \sin^{4}{\left(\frac{a - b}{2} \right)}}{\sin^{2}{\left(a - b \right)}} + 1}$$
2/a b\ 2/a b\
1 - tan |- - -| 1 - tan |- + -|
\2 2/ \2 2/
------------------- - -------------------
/ 2/a b\\ / 2/a b\\
2*|1 + tan |- - -|| 2*|1 + tan |- + -||
\ \2 2// \ \2 2//
$$\frac{- \tan^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} + 1}{2 \left(\tan^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} + 1\right)} - \frac{- \tan^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)} + 1}{2 \left(\tan^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)} + 1\right)}$$
4/a b\
4*sin |- - -|
\2 2/
/ pi\ / pi \ 1 - -------------
sin|a + b + --| sin|a + -- - b| 2
\ 2 / \ 2 / sin (a - b)
- --------------- - --------------- + -----------------
2 2 4/a b\
4*sin |- - -|
\2 2/
1 + -------------
2
sin (a - b)
$$- \frac{\sin{\left(a - b + \frac{\pi}{2} \right)}}{2} - \frac{\sin{\left(a + b + \frac{\pi}{2} \right)}}{2} + \frac{- \frac{4 \sin^{4}{\left(\frac{a}{2} - \frac{b}{2} \right)}}{\sin^{2}{\left(a - b \right)}} + 1}{\frac{4 \sin^{4}{\left(\frac{a}{2} - \frac{b}{2} \right)}}{\sin^{2}{\left(a - b \right)}} + 1}$$
2/a b\ / 2/a\\ / 2/b\\
-1 + cot |- - -| |-1 + cot |-||*|-1 + cot |-||
\2 2/ \ \2// \ \2//
---------------- - -----------------------------
2/a b\ / 2/a\\ / 2/b\\
1 + cot |- - -| |1 + cot |-||*|1 + cot |-||
\2 2/ \ \2// \ \2//
$$- \frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right) \left(\cot^{2}{\left(\frac{b}{2} \right)} - 1\right)}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{b}{2} \right)} + 1\right)} + \frac{\cot^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} + 1}$$
2/a pi b\
cos |- - -- - -|
\2 2 2/
1 - ----------------
2/a b\
cos |- - -|
cos(a + b) cos(a - b) \2 2/
- ---------- - ---------- + --------------------
2 2 2/a pi b\
cos |- - -- - -|
\2 2 2/
1 + ----------------
2/a b\
cos |- - -|
\2 2/
$$- \frac{\cos{\left(a - b \right)}}{2} - \frac{\cos{\left(a + b \right)}}{2} + \frac{1 - \frac{\cos^{2}{\left(\frac{a}{2} - \frac{b}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}}{1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{b}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}}$$
2/a b\
sec |- - -|
\2 2/
1 - ----------------
2/a pi b\
sec |- - -- - -|
1 1 \2 2 2/
- ------------ - ------------ + --------------------
2*sec(a + b) 2*sec(a - b) 2/a b\
sec |- - -|
\2 2/
1 + ----------------
2/a pi b\
sec |- - -- - -|
\2 2 2/
$$\frac{- \frac{\sec^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{b}{2} - \frac{\pi}{2} \right)}} + 1}{\frac{\sec^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{b}{2} - \frac{\pi}{2} \right)}} + 1} - \frac{1}{2 \sec{\left(a + b \right)}} - \frac{1}{2 \sec{\left(a - b \right)}}$$
2/a b\ / 2/a\\ / 2/b\\
1 - tan |- - -| |1 - tan |-||*|1 - tan |-||
\2 2/ \ \2// \ \2//
--------------- - ---------------------------
2/a b\ / 2/a\\ / 2/b\\
1 + tan |- - -| |1 + tan |-||*|1 + tan |-||
\2 2/ \ \2// \ \2//
$$- \frac{\left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(- \tan^{2}{\left(\frac{b}{2} \right)} + 1\right)}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{b}{2} \right)} + 1\right)} + \frac{- \tan^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} + 1}{\tan^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} + 1}$$
// 0 for a mod pi = 0\ // 0 for b mod pi = 0\ || | || | || /a\ | || /b\ | || 2*cot|-| | || 2*cot|-| | |< \2/ |*|< \2/ | ||----------- otherwise | ||----------- otherwise | || 2/a\ | || 2/b\ | ||1 + cot |-| | ||1 + cot |-| | \\ \2/ / \\ \2/ /
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{b}{2} \right)}}{\cot^{2}{\left(\frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
2/pi b a\
csc |-- + - - -|
\2 2 2/
1 - ----------------
2/a b\
csc |- - -|
1 1 \2 2/
- ----------------- - ----------------- + --------------------
/ pi \ /pi \ 2/pi b a\
2*csc|b + -- - a| 2*csc|-- - a - b| csc |-- + - - -|
\ 2 / \2 / \2 2 2/
1 + ----------------
2/a b\
csc |- - -|
\2 2/
$$\frac{1 - \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{b}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}}{1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{b}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}} - \frac{1}{2 \csc{\left(- a + b + \frac{\pi}{2} \right)}} - \frac{1}{2 \csc{\left(- a - b + \frac{\pi}{2} \right)}}$$
/a b pi\ /a pi\ /b pi\
2*tan|- - - + --| 4*tan|- + --|*tan|- + --|
\2 2 4 / \2 4 / \2 4 /
-------------------- - -------------------------------------
2/a b pi\ / 2/a pi\\ / 2/b pi\\
1 + tan |- - - + --| |1 + tan |- + --||*|1 + tan |- + --||
\2 2 4 / \ \2 4 // \ \2 4 //
$$\frac{2 \tan{\left(\frac{a}{2} - \frac{b}{2} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{a}{2} - \frac{b}{2} + \frac{\pi}{4} \right)} + 1} - \frac{4 \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)} \tan{\left(\frac{b}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\tan^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)} + 1\right)}$$
// 1 for a mod 2*pi = 0\ // 1 for b mod 2*pi = 0\ // 1 for (a - b) mod 2*pi = 0\ - |< |*|< | + |< | \\cos(a) otherwise / \\cos(b) otherwise / \\cos(a - b) otherwise /
$$\left(- \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\cos{\left(b \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \left(a - b\right) \bmod 2 \pi = 0 \\\cos{\left(a - b \right)} & \text{otherwise} \end{cases}\right)$$
/b a pi\ /a pi\ /b pi\
2*cot|- - - + --| 4*tan|- + --|*tan|- + --|
\2 2 4 / \2 4 / \2 4 /
-------------------- - -------------------------------------
2/b a pi\ / 2/a pi\\ / 2/b pi\\
1 + cot |- - - + --| |1 + tan |- + --||*|1 + tan |- + --||
\2 2 4 / \ \2 4 // \ \2 4 //
$$\frac{2 \cot{\left(- \frac{a}{2} + \frac{b}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(- \frac{a}{2} + \frac{b}{2} + \frac{\pi}{4} \right)} + 1} - \frac{4 \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)} \tan{\left(\frac{b}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\tan^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)} + 1\right)}$$
1 / 1 \ / 1 \
1 - ----------- |1 - -------|*|1 - -------|
2/a b\ | 2/a\| | 2/b\|
cot |- - -| | cot |-|| | cot |-||
\2 2/ \ \2// \ \2//
--------------- - ---------------------------
1 / 1 \ / 1 \
1 + ----------- |1 + -------|*|1 + -------|
2/a b\ | 2/a\| | 2/b\|
cot |- - -| | cot |-|| | cot |-||
\2 2/ \ \2// \ \2//
$$- \frac{\left(1 - \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \left(1 - \frac{1}{\cot^{2}{\left(\frac{b}{2} \right)}}\right)}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \left(1 + \frac{1}{\cot^{2}{\left(\frac{b}{2} \right)}}\right)} + \frac{1 - \frac{1}{\cot^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}}{1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}}$$
// 1 for a mod 2*pi = 0\ // 1 for b mod 2*pi = 0\ // 1 for (a - b) mod 2*pi = 0\ || | || | || | - |< 1 |*|< 1 | + |< 1 | ||------ otherwise | ||------ otherwise | ||---------- otherwise | \\sec(a) / \\sec(b) / \\sec(a - b) /
$$\left(- \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(b \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \left(a - b\right) \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(a - b \right)}} & \text{otherwise} \end{cases}\right)$$
/ 1 for (a + b) mod 2*pi = 0 / 1 for (a - b) mod 2*pi = 0 2/a b\
< < 1 - tan |- - -|
\cos(a + b) otherwise \cos(a - b) otherwise \2 2/
- ------------------------------------- - ------------------------------------- + ---------------
2 2 2/a b\
1 + tan |- - -|
\2 2/
$$\left(- \frac{\begin{cases} 1 & \text{for}\: \left(a - b\right) \bmod 2 \pi = 0 \\\cos{\left(a - b \right)} & \text{otherwise} \end{cases}}{2}\right) - \left(\frac{\begin{cases} 1 & \text{for}\: \left(a + b\right) \bmod 2 \pi = 0 \\\cos{\left(a + b \right)} & \text{otherwise} \end{cases}}{2}\right) + \frac{- \tan^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} + 1}{\tan^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} + 1}$$
// 1 for a mod 2*pi = 0\ // 1 for b mod 2*pi = 0\ // 1 for (a - b) mod 2*pi = 0\ || | || | || | - |< / pi\ |*|< / pi\ | + |< / pi \ | ||sin|a + --| otherwise | ||sin|b + --| otherwise | ||sin|a + -- - b| otherwise | \\ \ 2 / / \\ \ 2 / / \\ \ 2 / /
$$\left(- \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\sin{\left(a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\sin{\left(b + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \left(a - b\right) \bmod 2 \pi = 0 \\\sin{\left(a - b + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\ // 1 for b mod 2*pi = 0\ // 1 for (a - b) mod 2*pi = 0\ || | || | || | || 1 | || 1 | || 1 | - |<----------- otherwise |*|<----------- otherwise | + |<--------------- otherwise | || /pi \ | || /pi \ | || / pi \ | ||csc|-- - a| | ||csc|-- - b| | ||csc|b + -- - a| | \\ \2 / / \\ \2 / / \\ \ 2 / /
$$\left(- \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- b + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \left(a - b\right) \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- a + b + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// 1 for (a - b) mod 2*pi = 0\
|| |
|| 2/a b\ |
// 1 for a mod 2*pi = 0\ // 1 for b mod 2*pi = 0\ ||1 - tan |- - -| |
- |< |*|< | + |< \2 2/ |
\\cos(a) otherwise / \\cos(b) otherwise / ||--------------- otherwise |
|| 2/a b\ |
||1 + tan |- - -| |
\\ \2 2/ /
$$\left(- \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\cos{\left(b \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \left(a - b\right) \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} + 1}{\tan^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// 1 for (a - b) mod 2*pi = 0\
|| |
|| 1 |
||-1 + ----------- |
|| 2/a b\ |
// 1 for a mod 2*pi = 0\ // 1 for b mod 2*pi = 0\ || tan |- - -| |
- |< |*|< | + |< \2 2/ |
\\cos(a) otherwise / \\cos(b) otherwise / ||---------------- otherwise |
|| 1 |
||1 + ----------- |
|| 2/a b\ |
|| tan |- - -| |
\\ \2 2/ /
$$\left(- \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\cos{\left(b \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \left(a - b\right) \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}} & \text{otherwise} \end{cases}\right)$$
4/a - b\ / 4/a\\ / 4/b\\
4*sin |-----| | 4*sin |-|| | 4*sin |-||
\ 2 / | \2/| | \2/|
1 - ------------- |1 - ---------|*|1 - ---------|
2 | 2 | | 2 |
sin (a - b) \ sin (a) / \ sin (b) /
----------------- - -------------------------------
4/a - b\ / 4/a\\ / 4/b\\
4*sin |-----| | 4*sin |-|| | 4*sin |-||
\ 2 / | \2/| | \2/|
1 + ------------- |1 + ---------|*|1 + ---------|
2 | 2 | | 2 |
sin (a - b) \ sin (a) / \ sin (b) /
$$- \frac{\left(- \frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right) \left(- \frac{4 \sin^{4}{\left(\frac{b}{2} \right)}}{\sin^{2}{\left(b \right)}} + 1\right)}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right) \left(\frac{4 \sin^{4}{\left(\frac{b}{2} \right)}}{\sin^{2}{\left(b \right)}} + 1\right)} + \frac{- \frac{4 \sin^{4}{\left(\frac{a - b}{2} \right)}}{\sin^{2}{\left(a - b \right)}} + 1}{\frac{4 \sin^{4}{\left(\frac{a - b}{2} \right)}}{\sin^{2}{\left(a - b \right)}} + 1}$$
4/a b\ / 4/a\\ / 4/b\\
4*sin |- - -| | 4*sin |-|| | 4*sin |-||
\2 2/ | \2/| | \2/|
1 - ------------- |1 - ---------|*|1 - ---------|
2 | 2 | | 2 |
sin (a - b) \ sin (a) / \ sin (b) /
----------------- - -------------------------------
4/a b\ / 4/a\\ / 4/b\\
4*sin |- - -| | 4*sin |-|| | 4*sin |-||
\2 2/ | \2/| | \2/|
1 + ------------- |1 + ---------|*|1 + ---------|
2 | 2 | | 2 |
sin (a - b) \ sin (a) / \ sin (b) /
$$- \frac{\left(- \frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right) \left(- \frac{4 \sin^{4}{\left(\frac{b}{2} \right)}}{\sin^{2}{\left(b \right)}} + 1\right)}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right) \left(\frac{4 \sin^{4}{\left(\frac{b}{2} \right)}}{\sin^{2}{\left(b \right)}} + 1\right)} + \frac{- \frac{4 \sin^{4}{\left(\frac{a}{2} - \frac{b}{2} \right)}}{\sin^{2}{\left(a - b \right)}} + 1}{\frac{4 \sin^{4}{\left(\frac{a}{2} - \frac{b}{2} \right)}}{\sin^{2}{\left(a - b \right)}} + 1}$$
// 1 for a mod 2*pi = 0\ // 1 for b mod 2*pi = 0\ // 1 for (a - b) mod 2*pi = 0\ || | || | || | || 2/a\ | || 2/b\ | || 2/a b\ | ||-1 + cot |-| | ||-1 + cot |-| | ||-1 + 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/ /
$$\left(- \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{b}{2} \right)} - 1}{\cot^{2}{\left(\frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \left(a - b\right) \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\ // 1 for b mod 2*pi = 0\ // 1 for (a - b) mod 2*pi = 0\ || | || | || | || 2/a\ | || 2/b\ | || 2/a b\ | ||1 - tan |-| | ||1 - tan |-| | ||1 - 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/ /
$$\left(- \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{a}{2} \right)} + 1}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{b}{2} \right)} + 1}{\tan^{2}{\left(\frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \left(a - b\right) \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} + 1}{\tan^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
/ 1 for (a + b) mod 2*pi = 0 / 1 for (a - b) mod 2*pi = 0
| |
| 2/a b\ | 2/a b\
|-1 + cot |- + -| |-1 + cot |- - -|
< \2 2/ < \2 2/ 1
|---------------- otherwise |---------------- otherwise 1 - -----------
| 2/a b\ | 2/a b\ 2/a b\
|1 + cot |- + -| |1 + cot |- - -| cot |- - -|
\ \2 2/ \ \2 2/ \2 2/
- ------------------------------------------- - ------------------------------------------- + ---------------
2 2 1
1 + -----------
2/a b\
cot |- - -|
\2 2/
$$\left(- \frac{\begin{cases} 1 & \text{for}\: \left(a - b\right) \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases}}{2}\right) - \left(\frac{\begin{cases} 1 & \text{for}\: \left(a + b\right) \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} + \frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases}}{2}\right) + \frac{1 - \frac{1}{\cot^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}}{1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}}$$
// 1 for a mod 2*pi = 0\ // 1 for b mod 2*pi = 0\ // 1 for (a - b) mod 2*pi = 0\
|| | || | || |
- |
$$\left(- \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\cos{\left(b \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \left(a - b\right) \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \left(a - b\right) \bmod 2 \pi = 0 \\\cos{\left(a - b \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
// / pi\ \ // / pi\ \ // / pi \ \ || 0 for |a + --| mod pi = 0| || 0 for |b + --| mod pi = 0| || 0 for |a + -- - b| mod pi = 0| || \ 2 / | || \ 2 / | || \ 2 / | - |< |*|< | + |< | || /a pi\ | || /b pi\ | || /a b pi\ | ||(1 + sin(a))*cot|- + --| otherwise | ||(1 + sin(b))*cot|- + --| otherwise | ||(1 + sin(a - b))*cot|- - - + --| otherwise | \\ \2 4 / / \\ \2 4 / / \\ \2 2 4 / /
$$\left(- \left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(a \right)} + 1\right) \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(b + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(b \right)} + 1\right) \cot{\left(\frac{b}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \left(a - b + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(a - b \right)} + 1\right) \cot{\left(\frac{a}{2} - \frac{b}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\ // 1 for b mod 2*pi = 0\ // 1 for (a - b) mod 2*pi = 0\ || | || | || | || 1 | || 1 | || 1 | ||-1 + ------- | ||-1 + ------- | ||-1 + ----------- | || 2/a\ | || 2/b\ | || 2/a b\ | || tan |-| | || tan |-| | || tan |- - -| | - |< \2/ |*|< \2/ | + |< \2 2/ | ||------------ otherwise | ||------------ otherwise | ||---------------- otherwise | || 1 | || 1 | || 1 | ||1 + ------- | ||1 + ------- | ||1 + ----------- | || 2/a\ | || 2/b\ | || 2/a b\ | || tan |-| | || tan |-| | || tan |- - -| | \\ \2/ / \\ \2/ / \\ \2 2/ /
$$\left(- \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{b}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{b}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \left(a - b\right) \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}} & \text{otherwise} \end{cases}\right)$$
2/a pi b\ / 2/a pi\\ / 2/b pi\\
cos |- - -- - -| | cos |- - --|| | cos |- - --||
\2 2 2/ | \2 2 /| | \2 2 /|
1 - ---------------- |1 - ------------|*|1 - ------------|
2/a b\ | 2/a\ | | 2/b\ |
cos |- - -| | cos |-| | | cos |-| |
\2 2/ \ \2/ / \ \2/ /
-------------------- - -------------------------------------
2/a pi b\ / 2/a pi\\ / 2/b pi\\
cos |- - -- - -| | cos |- - --|| | cos |- - --||
\2 2 2/ | \2 2 /| | \2 2 /|
1 + ---------------- |1 + ------------|*|1 + ------------|
2/a b\ | 2/a\ | | 2/b\ |
cos |- - -| | cos |-| | | cos |-| |
\2 2/ \ \2/ / \ \2/ /
$$- \frac{\left(1 - \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right) \left(1 - \frac{\cos^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{b}{2} \right)}}\right)}{\left(1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right) \left(1 + \frac{\cos^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{b}{2} \right)}}\right)} + \frac{1 - \frac{\cos^{2}{\left(\frac{a}{2} - \frac{b}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}}{1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{b}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}}$$
2/a b\ / 2/a\ \ / 2/b\ \
sec |- - -| | sec |-| | | sec |-| |
\2 2/ | \2/ | | \2/ |
1 - ---------------- |1 - ------------|*|1 - ------------|
2/a pi b\ | 2/a pi\| | 2/b pi\|
sec |- - -- - -| | sec |- - --|| | sec |- - --||
\2 2 2/ \ \2 2 // \ \2 2 //
-------------------- - -------------------------------------
2/a b\ / 2/a\ \ / 2/b\ \
sec |- - -| | sec |-| | | sec |-| |
\2 2/ | \2/ | | \2/ |
1 + ---------------- |1 + ------------|*|1 + ------------|
2/a pi b\ | 2/a pi\| | 2/b pi\|
sec |- - -- - -| | sec |- - --|| | sec |- - --||
\2 2 2/ \ \2 2 // \ \2 2 //
$$- \frac{\left(- \frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \left(- \frac{\sec^{2}{\left(\frac{b}{2} \right)}}{\sec^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}} + 1\right)}{\left(\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \left(\frac{\sec^{2}{\left(\frac{b}{2} \right)}}{\sec^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}} + 1\right)} + \frac{- \frac{\sec^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{b}{2} - \frac{\pi}{2} \right)}} + 1}{\frac{\sec^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{b}{2} - \frac{\pi}{2} \right)}} + 1}$$
2/pi b a\ / 2/pi a\\ / 2/pi b\\
csc |-- + - - -| | csc |-- - -|| | csc |-- - -||
\2 2 2/ | \2 2/| | \2 2/|
1 - ---------------- |1 - ------------|*|1 - ------------|
2/a b\ | 2/a\ | | 2/b\ |
csc |- - -| | csc |-| | | csc |-| |
\2 2/ \ \2/ / \ \2/ /
-------------------- - -------------------------------------
2/pi b a\ / 2/pi a\\ / 2/pi b\\
csc |-- + - - -| | csc |-- - -|| | csc |-- - -||
\2 2 2/ | \2 2/| | \2 2/|
1 + ---------------- |1 + ------------|*|1 + ------------|
2/a b\ | 2/a\ | | 2/b\ |
csc |- - -| | csc |-| | | csc |-| |
\2 2/ \ \2/ / \ \2/ /
$$- \frac{\left(1 - \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right) \left(1 - \frac{\csc^{2}{\left(- \frac{b}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{b}{2} \right)}}\right)}{\left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right) \left(1 + \frac{\csc^{2}{\left(- \frac{b}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{b}{2} \right)}}\right)} + \frac{1 - \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{b}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}}{1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{b}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}}$$
// / pi\ \ // / pi\ \ // / pi \ \ || 0 for |a + --| mod pi = 0| || 0 for |b + --| mod pi = 0| || 0 for |a + -- - b| mod pi = 0| || \ 2 / | || \ 2 / | || \ 2 / | || | || | || | || /a pi\ | || /b pi\ | || /a b pi\ | - |< 2*cot|- + --| |*|< 2*cot|- + --| | + |< 2*cot|- - - + --| | || \2 4 / | || \2 4 / | || \2 2 4 / | ||---------------- otherwise | ||---------------- otherwise | ||-------------------- otherwise | || 2/a pi\ | || 2/b pi\ | || 2/a b pi\ | ||1 + cot |- + --| | ||1 + cot |- + --| | ||1 + cot |- - - + --| | \\ \2 4 / / \\ \2 4 / / \\ \2 2 4 / /
$$\left(- \left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(b + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{b}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \left(a - b + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} - \frac{b}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{a}{2} - \frac{b}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// 1 for (a - b) mod 2*pi = 0\
// 1 for a mod 2*pi = 0\ // 1 for b mod 2*pi = 0\ || |
|| | || | || 2 4/a b\ |
|| 2 | || 2 | ||sin (a - b) - 4*sin |- - -| |
- |< -4 + 4*sin (a) + 4*cos(a) |*|< -4 + 4*sin (b) + 4*cos(b) | + |< \2 2/ |
||--------------------------- otherwise | ||--------------------------- otherwise | ||--------------------------- otherwise |
|| 2 2 | || 2 2 | || 2 4/a b\ |
\\2*(1 - cos(a)) + 2*sin (a) / \\2*(1 - cos(b)) + 2*sin (b) / ||sin (a - b) + 4*sin |- - -| |
\\ \2 2/ /
$$\left(- \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{4 \sin^{2}{\left(a \right)} + 4 \cos{\left(a \right)} - 4}{2 \left(- \cos{\left(a \right)} + 1\right)^{2} + 2 \sin^{2}{\left(a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{4 \sin^{2}{\left(b \right)} + 4 \cos{\left(b \right)} - 4}{2 \left(- \cos{\left(b \right)} + 1\right)^{2} + 2 \sin^{2}{\left(b \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \left(a - b\right) \bmod 2 \pi = 0 \\\frac{- 4 \sin^{4}{\left(\frac{a}{2} - \frac{b}{2} \right)} + \sin^{2}{\left(a - b \right)}}{4 \sin^{4}{\left(\frac{a}{2} - \frac{b}{2} \right)} + \sin^{2}{\left(a - b \right)}} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\ // 1 for b mod 2*pi = 0\ // 1 for (a - b) mod 2*pi = 0\ || | || | || | || 2 | || 2 | || 2 | || sin (a) | || sin (b) | || sin (a - b) | ||-1 + --------- | ||-1 + --------- | ||-1 + ------------- | || 4/a\ | || 4/b\ | || 4/a b\ | || 4*sin |-| | || 4*sin |-| | || 4*sin |- - -| | - |< \2/ |*|< \2/ | + |< \2 2/ | ||-------------- otherwise | ||-------------- otherwise | ||------------------ otherwise | || 2 | || 2 | || 2 | || sin (a) | || sin (b) | || sin (a - b) | ||1 + --------- | ||1 + --------- | ||1 + ------------- | || 4/a\ | || 4/b\ | || 4/a b\ | || 4*sin |-| | || 4*sin |-| | || 4*sin |- - -| | \\ \2/ / \\ \2/ / \\ \2 2/ /
$$\left(- \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}}{1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(b \right)}}{4 \sin^{4}{\left(\frac{b}{2} \right)}}}{1 + \frac{\sin^{2}{\left(b \right)}}{4 \sin^{4}{\left(\frac{b}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \left(a - b\right) \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(a - b \right)}}{4 \sin^{4}{\left(\frac{a}{2} - \frac{b}{2} \right)}}}{1 + \frac{\sin^{2}{\left(a - b \right)}}{4 \sin^{4}{\left(\frac{a}{2} - \frac{b}{2} \right)}}} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\ // 1 for b mod 2*pi = 0\ // 1 for (a - b) mod 2*pi = 0\ || | || | || | ||/ 1 for a mod 2*pi = 0 | ||/ 1 for b mod 2*pi = 0 | ||/ 1 for (a - b) mod 2*pi = 0 | ||| | ||| | ||| | ||| 2/a\ | ||| 2/b\ | ||| 2/a b\ | - |<|-1 + cot |-| |*|<|-1 + cot |-| | + |<|-1 + 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/ /
$$\left(- \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{b}{2} \right)} - 1}{\cot^{2}{\left(\frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \left(a - b\right) \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \left(a - b\right) \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\ // 1 for b mod 2*pi = 0\ // 1 for (a - b) mod 2*pi = 0\ || | || | || | || 2/a\ | || 2/b\ | || 2/a b\ | || cos |-| | || cos |-| | || cos |- - -| | || \2/ | || \2/ | || \2 2/ | ||-1 + ------------ | ||-1 + ------------ | ||-1 + ---------------- | || 2/a pi\ | || 2/b pi\ | || 2/a pi b\ | || cos |- - --| | || cos |- - --| | || cos |- - -- - -| | - |< \2 2 / |*|< \2 2 / | + |< \2 2 2/ | ||----------------- otherwise | ||----------------- otherwise | ||--------------------- otherwise | || 2/a\ | || 2/b\ | || 2/a b\ | || cos |-| | || cos |-| | || cos |- - -| | || \2/ | || \2/ | || \2 2/ | || 1 + ------------ | || 1 + ------------ | || 1 + ---------------- | || 2/a pi\ | || 2/b pi\ | || 2/a pi b\ | || cos |- - --| | || cos |- - --| | || cos |- - -- - -| | \\ \2 2 / / \\ \2 2 / / \\ \2 2 2/ /
$$\left(- \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{b}{2} \right)}}{\cos^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{b}{2} \right)}}{\cos^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \left(a - b\right) \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{b}{2} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{b}{2} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\ // 1 for b mod 2*pi = 0\ // 1 for (a - b) mod 2*pi = 0\ || | || | || | || 2/a pi\ | || 2/b pi\ | || 2/a pi b\ | || sec |- - --| | || sec |- - --| | || sec |- - -- - -| | || \2 2 / | || \2 2 / | || \2 2 2/ | ||-1 + ------------ | ||-1 + ------------ | ||-1 + ---------------- | || 2/a\ | || 2/b\ | || 2/a b\ | || sec |-| | || sec |-| | || sec |- - -| | - |< \2/ |*|< \2/ | + |< \2 2/ | ||----------------- otherwise | ||----------------- otherwise | ||--------------------- otherwise | || 2/a pi\ | || 2/b pi\ | || 2/a pi b\ | || sec |- - --| | || sec |- - --| | || sec |- - -- - -| | || \2 2 / | || \2 2 / | || \2 2 2/ | || 1 + ------------ | || 1 + ------------ | || 1 + ---------------- | || 2/a\ | || 2/b\ | || 2/a b\ | || sec |-| | || sec |-| | || sec |- - -| | \\ \2/ / \\ \2/ / \\ \2 2/ /
$$\left(- \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{b}{2} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{b}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \left(a - b\right) \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{b}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{b}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\ // 1 for b mod 2*pi = 0\ // 1 for (a - b) mod 2*pi = 0\ || | || | || | || 2/a\ | || 2/b\ | || 2/a b\ | || csc |-| | || csc |-| | || csc |- - -| | || \2/ | || \2/ | || \2 2/ | ||-1 + ------------ | ||-1 + ------------ | ||-1 + ---------------- | || 2/pi a\ | || 2/pi b\ | || 2/pi b a\ | || csc |-- - -| | || csc |-- - -| | || csc |-- + - - -| | - |< \2 2/ |*|< \2 2/ | + |< \2 2 2/ | ||----------------- otherwise | ||----------------- otherwise | ||--------------------- otherwise | || 2/a\ | || 2/b\ | || 2/a b\ | || csc |-| | || csc |-| | || csc |- - -| | || \2/ | || \2/ | || \2 2/ | || 1 + ------------ | || 1 + ------------ | || 1 + ---------------- | || 2/pi a\ | || 2/pi b\ | || 2/pi b a\ | || csc |-- - -| | || csc |-- - -| | || csc |-- + - - -| | \\ \2 2/ / \\ \2 2/ / \\ \2 2 2/ /
$$\left(- \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{b}{2} \right)}}{\csc^{2}{\left(- \frac{b}{2} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{b}{2} \right)}}{\csc^{2}{\left(- \frac{b}{2} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \left(a - b\right) \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{b}{2} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{b}{2} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)$$
-Piecewise((1, Mod(a = 2*pi, 0)), ((-1 + csc(a/2)^2/csc(pi/2 - a/2)^2)/(1 + csc(a/2)^2/csc(pi/2 - a/2)^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((1, Mod(a - b = 2*pi, 0)), ((-1 + csc(a/2 - b/2)^2/csc(pi/2 + b/2 - a/2)^2)/(1 + csc(a/2 - b/2)^2/csc(pi/2 + b/2 - a/2)^2), True))