Господин Экзамен
17*(cos(a)*cos(b)+sin(a)*sin(b))+sin(b) если a=3
Решение
Вы ввели
[src]
17*(cos(a)*cos(b) + sin(a)*sin(b)) + sin(b)
$$17 \left(\sin{\left(a \right)} \sin{\left(b \right)} + \cos{\left(a \right)} \cos{\left(b \right)}\right) + \sin{\left(b \right)}$$
17*(cos(a)*cos(b) + sin(a)*sin(b)) + sin(b)
Общее упрощение
[src]
17*cos(a - b) + sin(b)
$$\sin{\left(b \right)} + 17 \cos{\left(a - b \right)}$$
17*cos(a - b) + sin(b)
Подстановка условия
[src]
17*(cos(a)*cos(b) + sin(a)*sin(b)) + sin(b) при a = 3
подставляем
17*(cos(a)*cos(b) + sin(a)*sin(b)) + sin(b)
$$17 \left(\sin{\left(a \right)} \sin{\left(b \right)} + \cos{\left(a \right)} \cos{\left(b \right)}\right) + \sin{\left(b \right)}$$
17*cos(a - b) + sin(b)
$$\sin{\left(b \right)} + 17 \cos{\left(a - b \right)}$$
переменные
a = 3
$$a = 3$$
17*cos((3) - b) + sin(b)
$$\sin{\left(b \right)} + 17 \cos{\left((3) - b \right)}$$
17*cos(3 - b) + sin(b)
$$\sin{\left(b \right)} + 17 \cos{\left(- b + 3 \right)}$$
17*cos(-3 + b) + sin(b)
$$\sin{\left(b \right)} + 17 \cos{\left(b - 3 \right)}$$
17*cos(-3 + b) + sin(b)
Собрать выражение
[src]
17*cos(a - b) + sin(b)
$$\sin{\left(b \right)} + 17 \cos{\left(a - b \right)}$$
17*cos(a)*cos(b) + 17*sin(a)*sin(b) + sin(b)
$$17 \sin{\left(a \right)} \sin{\left(b \right)} + 17 \cos{\left(a \right)} \cos{\left(b \right)} + \sin{\left(b \right)}$$
17*cos(a)*cos(b) + 17*sin(a)*sin(b) + sin(b)
Рациональный знаменатель
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17*cos(a)*cos(b) + 17*sin(a)*sin(b) + sin(b)
$$17 \sin{\left(a \right)} \sin{\left(b \right)} + 17 \cos{\left(a \right)} \cos{\left(b \right)} + \sin{\left(b \right)}$$
17*cos(a)*cos(b) + 17*sin(a)*sin(b) + sin(b)
Степени
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17*cos(a)*cos(b) + 17*sin(a)*sin(b) + sin(b)
$$17 \sin{\left(a \right)} \sin{\left(b \right)} + 17 \cos{\left(a \right)} \cos{\left(b \right)} + \sin{\left(b \right)}$$
/ I*a -I*a\ / I*b -I*b\ / -I*a I*a\ / -I*b I*b\ / -I*b I*b\ |e e | |e e | 17*\- e + e /*\- e + e / I*\- e + e / 17*|---- + -----|*|---- + -----| - ------------------------------------ - ------------------ \ 2 2 / \ 2 2 / 4 2
$$17 \left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2}\right) \left(\frac{e^{i b}}{2} + \frac{e^{- i b}}{2}\right) - \frac{17 \left(e^{i a} - e^{- i a}\right) \left(e^{i b} - e^{- i b}\right)}{4} - \frac{i \left(e^{i b} - e^{- i b}\right)}{2}$$
17*(exp(i*a)/2 + exp(-i*a)/2)*(exp(i*b)/2 + exp(-i*b)/2) - 17*(-exp(-i*a) + exp(i*a))*(-exp(-i*b) + exp(i*b))/4 - i*(-exp(-i*b) + exp(i*b))/2
Численный ответ
[src]
17.0*cos(a)*cos(b) + 17.0*sin(a)*sin(b) + sin(b)
17.0*cos(a)*cos(b) + 17.0*sin(a)*sin(b) + sin(b)
Комбинаторика
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17*cos(a)*cos(b) + 17*sin(a)*sin(b) + sin(b)
$$17 \sin{\left(a \right)} \sin{\left(b \right)} + 17 \cos{\left(a \right)} \cos{\left(b \right)} + \sin{\left(b \right)}$$
17*cos(a)*cos(b) + 17*sin(a)*sin(b) + sin(b)
Общий знаменатель
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17*cos(a)*cos(b) + 17*sin(a)*sin(b) + sin(b)
$$17 \sin{\left(a \right)} \sin{\left(b \right)} + 17 \cos{\left(a \right)} \cos{\left(b \right)} + \sin{\left(b \right)}$$
17*cos(a)*cos(b) + 17*sin(a)*sin(b) + sin(b)
Тригонометрическая часть
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17*cos(a - b) + sin(b)
$$\sin{\left(b \right)} + 17 \cos{\left(a - b \right)}$$
/ pi\
17*cos(a - b) + cos|b - --|
\ 2 /
$$17 \cos{\left(a - b \right)} + \cos{\left(b - \frac{\pi}{2} \right)}$$
/ pi \
17*sin|a + -- - b| + sin(b)
\ 2 /
$$\sin{\left(b \right)} + 17 \sin{\left(a - b + \frac{\pi}{2} \right)}$$
1 17 ----------- + ---------- / pi\ sec(a - b) sec|b - --| \ 2 /
$$\frac{1}{\sec{\left(b - \frac{\pi}{2} \right)}} + \frac{17}{\sec{\left(a - b \right)}}$$
1 17
------ + ---------------
csc(b) / pi \
csc|b + -- - a|
\ 2 /
$$\frac{17}{\csc{\left(- a + b + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(b \right)}}$$
17*cos(a)*cos(b) + 17*sin(a)*sin(b) + sin(b)
$$17 \sin{\left(a \right)} \sin{\left(b \right)} + 17 \cos{\left(a \right)} \cos{\left(b \right)} + \sin{\left(b \right)}$$
1 17 17 ------ + ------------- + ------------- csc(b) csc(a)*csc(b) sec(a)*sec(b)
$$\frac{1}{\csc{\left(b \right)}} + \frac{17}{\sec{\left(a \right)} \sec{\left(b \right)}} + \frac{17}{\csc{\left(a \right)} \csc{\left(b \right)}}$$
/ pi\ / pi\
17*sin(a)*sin(b) + 17*sin|a + --|*sin|b + --| + sin(b)
\ 2 / \ 2 /
$$17 \sin{\left(a \right)} \sin{\left(b \right)} + 17 \sin{\left(a + \frac{\pi}{2} \right)} \sin{\left(b + \frac{\pi}{2} \right)} + \sin{\left(b \right)}$$
/ 2/a b\\
17*|1 - tan |- - -||
\ \2 2//
-------------------- + sin(b)
2/a b\
1 + tan |- - -|
\2 2/
$$\sin{\left(b \right)} + \frac{17 \cdot \left(- \tan^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} + 1\right)}{\tan^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} + 1}$$
/ pi\ / pi\ / pi\
17*cos(a)*cos(b) + 17*cos|a - --|*cos|b - --| + cos|b - --|
\ 2 / \ 2 / \ 2 /
$$17 \cos{\left(a \right)} \cos{\left(b \right)} + 17 \cos{\left(a - \frac{\pi}{2} \right)} \cos{\left(b - \frac{\pi}{2} \right)} + \cos{\left(b - \frac{\pi}{2} \right)}$$
1 17 17
------ + ------------- + -----------------------
csc(b) csc(a)*csc(b) /pi \ /pi \
csc|-- - a|*csc|-- - b|
\2 / \2 /
$$\frac{1}{\csc{\left(b \right)}} + \frac{17}{\csc{\left(- a + \frac{\pi}{2} \right)} \csc{\left(- b + \frac{\pi}{2} \right)}} + \frac{17}{\csc{\left(a \right)} \csc{\left(b \right)}}$$
1 17 17 ----------- + ------------- + ----------------------- / pi\ sec(a)*sec(b) / pi\ / pi\ sec|b - --| sec|a - --|*sec|b - --| \ 2 / \ 2 / \ 2 /
$$\frac{1}{\sec{\left(b - \frac{\pi}{2} \right)}} + \frac{17}{\sec{\left(a - \frac{\pi}{2} \right)} \sec{\left(b - \frac{\pi}{2} \right)}} + \frac{17}{\sec{\left(a \right)} \sec{\left(b \right)}}$$
1 17 17 ----------- + ------------- + ----------------------- /pi \ sec(a)*sec(b) /pi \ /pi \ sec|-- - b| sec|-- - a|*sec|-- - b| \2 / \2 / \2 /
$$\frac{1}{\sec{\left(- b + \frac{\pi}{2} \right)}} + \frac{17}{\sec{\left(- a + \frac{\pi}{2} \right)} \sec{\left(- b + \frac{\pi}{2} \right)}} + \frac{17}{\sec{\left(a \right)} \sec{\left(b \right)}}$$
1 17 17
----------- + ----------------------- + -----------------------
csc(pi - b) csc(pi - a)*csc(pi - b) /pi \ /pi \
csc|-- - a|*csc|-- - b|
\2 / \2 /
$$\frac{1}{\csc{\left(- b + \pi \right)}} + \frac{17}{\csc{\left(- a + \frac{\pi}{2} \right)} \csc{\left(- b + \frac{\pi}{2} \right)}} + \frac{17}{\csc{\left(- a + \pi \right)} \csc{\left(- b + \pi \right)}}$$
/b\ / 2/a b\\
2*tan|-| 17*|1 - tan |- - -||
\2/ \ \2 2//
----------- + --------------------
2/b\ 2/a b\
1 + tan |-| 1 + tan |- - -|
\2/ \2 2/
$$\frac{17 \cdot \left(- \tan^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} + 1\right)}{\tan^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} + 1} + \frac{2 \tan{\left(\frac{b}{2} \right)}}{\tan^{2}{\left(\frac{b}{2} \right)} + 1}$$
// 1 for (a - b) mod 2*pi = 0\ // 0 for b mod pi = 0\ 17*|< | + |< | \\cos(a - b) otherwise / \\sin(b) otherwise /
$$\left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\sin{\left(b \right)} & \text{otherwise} \end{cases}\right) + \left(17 \left(\begin{cases} 1 & \text{for}\: \left(a - b\right) \bmod 2 \pi = 0 \\\cos{\left(a - b \right)} & \text{otherwise} \end{cases}\right)\right)$$
/b\ /a\ /b\
(1 + cos(b))*tan|-| + 17*cos(a)*cos(b) + 17*(1 + cos(a)*cos(b) + cos(a) + cos(b))*tan|-|*tan|-|
\2/ \2/ \2/
$$17 \left(\cos{\left(a \right)} \cos{\left(b \right)} + \cos{\left(a \right)} + \cos{\left(b \right)} + 1\right) \tan{\left(\frac{a}{2} \right)} \tan{\left(\frac{b}{2} \right)} + \left(\cos{\left(b \right)} + 1\right) \tan{\left(\frac{b}{2} \right)} + 17 \cos{\left(a \right)} \cos{\left(b \right)}$$
// 1 for (a - b) mod 2*pi = 0\ // 0 for b mod pi = 0\ || | || | || 2/a b\ | || /b\ | ||-1 + cot |- - -| | || 2*cot|-| | 17*|< \2 2/ | + |< \2/ | ||---------------- otherwise | ||----------- otherwise | || 2/a b\ | || 2/b\ | ||1 + cot |- - -| | ||1 + cot |-| | \\ \2 2/ / \\ \2/ /
$$\left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{b}{2} \right)}}{\cot^{2}{\left(\frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(17 \left(\begin{cases} 1 & \text{for}\: \left(a - b\right) \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
/ 2/b pi\\ / 2/a pi\\ / 2/b pi\\
|1 - cot |- + --||*(1 + sin(b)) 17*|1 - cot |- + --||*|1 - cot |- + --||*(1 + sin(a))*(1 + sin(b))
17*(cos(a + b) + cos(a - b)) \ \2 4 // \ \2 4 // \ \2 4 //
---------------------------- + ------------------------------- + ------------------------------------------------------------------
2 2 4
$$\frac{17 \cdot \left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(- \cot^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(a \right)} + 1\right) \left(\sin{\left(b \right)} + 1\right)}{4} + \frac{\left(- \cot^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(b \right)} + 1\right)}{2} + \frac{17 \left(\cos{\left(a - b \right)} + \cos{\left(a + b \right)}\right)}{2}$$
/b\ / 2/a\\ / 2/b\\ /a\ /b\
2*tan|-| 17*|1 - tan |-||*|1 - tan |-|| 68*tan|-|*tan|-|
\2/ \ \2// \ \2// \2/ \2/
----------- + ------------------------------ + ---------------------------
2/b\ / 2/a\\ / 2/b\\ / 2/a\\ / 2/b\\
1 + tan |-| |1 + tan |-||*|1 + tan |-|| |1 + tan |-||*|1 + tan |-||
\2/ \ \2// \ \2// \ \2// \ \2//
$$\frac{17 \cdot \left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(- \tan^{2}{\left(\frac{b}{2} \right)} + 1\right)}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{b}{2} \right)} + 1\right)} + \frac{2 \tan{\left(\frac{b}{2} \right)}}{\tan^{2}{\left(\frac{b}{2} \right)} + 1} + \frac{68 \tan{\left(\frac{a}{2} \right)} \tan{\left(\frac{b}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{b}{2} \right)} + 1\right)}$$
/b\ /a\ /b\ /a pi\ /b pi\
2*cot|-| 68*cot|-|*cot|-| 68*tan|- + --|*tan|- + --|
\2/ \2/ \2/ \2 4 / \2 4 /
----------- + --------------------------- + -------------------------------------
2/b\ / 2/a\\ / 2/b\\ / 2/a pi\\ / 2/b pi\\
1 + cot |-| |1 + cot |-||*|1 + cot |-|| |1 + tan |- + --||*|1 + tan |- + --||
\2/ \ \2// \ \2// \ \2 4 // \ \2 4 //
$$\frac{2 \cot{\left(\frac{b}{2} \right)}}{\cot^{2}{\left(\frac{b}{2} \right)} + 1} + \frac{68 \cot{\left(\frac{a}{2} \right)} \cot{\left(\frac{b}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{b}{2} \right)} + 1\right)} + \frac{68 \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)} \tan{\left(\frac{b}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\tan^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)} + 1\right)}$$
/b\ /a\ /b\ /a pi\ /b pi\
2*tan|-| 68*tan|-|*tan|-| 68*tan|- + --|*tan|- + --|
\2/ \2/ \2/ \2 4 / \2 4 /
----------- + --------------------------- + -------------------------------------
2/b\ / 2/a\\ / 2/b\\ / 2/a pi\\ / 2/b pi\\
1 + tan |-| |1 + tan |-||*|1 + tan |-|| |1 + tan |- + --||*|1 + tan |- + --||
\2/ \ \2// \ \2// \ \2 4 // \ \2 4 //
$$\frac{68 \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)} \tan{\left(\frac{b}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\tan^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)} + 1\right)} + \frac{2 \tan{\left(\frac{b}{2} \right)}}{\tan^{2}{\left(\frac{b}{2} \right)} + 1} + \frac{68 \tan{\left(\frac{a}{2} \right)} \tan{\left(\frac{b}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{b}{2} \right)} + 1\right)}$$
/ 1 \ / 1 \
17*|1 - -------|*|1 - -------|
| 2/a\| | 2/b\|
| cot |-|| | cot |-||
2 \ \2// \ \2// 68
-------------------- + ------------------------------ + -----------------------------------------
/ 1 \ /b\ / 1 \ / 1 \ / 1 \ / 1 \ /a\ /b\
|1 + -------|*cot|-| |1 + -------|*|1 + -------| |1 + -------|*|1 + -------|*cot|-|*cot|-|
| 2/b\| \2/ | 2/a\| | 2/b\| | 2/a\| | 2/b\| \2/ \2/
| cot |-|| | cot |-|| | cot |-|| | cot |-|| | cot |-||
\ \2// \ \2// \ \2// \ \2// \ \2//
$$\frac{17 \cdot \left(1 - \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \left(1 - \frac{1}{\cot^{2}{\left(\frac{b}{2} \right)}}\right)}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \left(1 + \frac{1}{\cot^{2}{\left(\frac{b}{2} \right)}}\right)} + \frac{2}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{b}{2} \right)}}\right) \cot{\left(\frac{b}{2} \right)}} + \frac{68}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \left(1 + \frac{1}{\cot^{2}{\left(\frac{b}{2} \right)}}\right) \cot{\left(\frac{a}{2} \right)} \cot{\left(\frac{b}{2} \right)}}$$
// 0 for a mod pi = 0\ // 0 for b mod pi = 0\ // 1 for a mod 2*pi = 0\ // 1 for b mod 2*pi = 0\ // 0 for b mod pi = 0\ 17*|< |*|< | + 17*|< |*|< | + |< | \\sin(a) otherwise / \\sin(b) otherwise / \\cos(a) otherwise / \\cos(b) otherwise / \\sin(b) otherwise /
$$\left(17 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\sin{\left(b \right)} & \text{otherwise} \end{cases}\right)\right) + \left(17 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\cos{\left(b \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\sin{\left(b \right)} & \text{otherwise} \end{cases}\right)$$
2/b pi\ / 2/a\\ / 2/b\\ / 2/a pi\\ / 2/b pi\\
-1 + tan |- + --| 17*|-1 + cot |-||*|-1 + cot |-|| 17*|-1 + tan |- + --||*|-1 + tan |- + --||
\2 4 / \ \2// \ \2// \ \2 4 // \ \2 4 //
----------------- + -------------------------------- + ------------------------------------------
2/b pi\ / 2/a\\ / 2/b\\ / 2/a pi\\ / 2/b pi\\
1 + tan |- + --| |1 + cot |-||*|1 + cot |-|| |1 + tan |- + --||*|1 + tan |- + --||
\2 4 / \ \2// \ \2// \ \2 4 // \ \2 4 //
$$\frac{17 \left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1\right) \left(\tan^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)} - 1\right)}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\tan^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)} + 1\right)} + \frac{\tan^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)} + 1} + \frac{17 \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right) \left(\cot^{2}{\left(\frac{b}{2} \right)} - 1\right)}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{b}{2} \right)} + 1\right)}$$
// 1 for a mod 2*pi = 0\ // 1 for b mod 2*pi = 0\
// 0 for a mod pi = 0\ // 0 for b mod pi = 0\ || | || | // 0 for b mod pi = 0\
17*|< |*|< | + 17*|< / pi\ |*|< / pi\ | + |< |
\\sin(a) otherwise / \\sin(b) otherwise / ||sin|a + --| otherwise | ||sin|b + --| otherwise | \\sin(b) otherwise /
\\ \ 2 / / \\ \ 2 / /
$$\left(17 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\sin{\left(b \right)} & \text{otherwise} \end{cases}\right)\right) + \left(17 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\sin{\left(a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\sin{\left(b + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\sin{\left(b \right)} & \text{otherwise} \end{cases}\right)$$
2/b pi\ / 2/a pi\\ / 2/b pi\\ / 2/a\\ / 2/b\\
1 - cot |- + --| 17*|1 - cot |- + --||*|1 - cot |- + --|| 17*|1 - tan |-||*|1 - tan |-||
\2 4 / \ \2 4 // \ \2 4 // \ \2// \ \2//
---------------- + ---------------------------------------- + ------------------------------
2/b pi\ / 2/a pi\\ / 2/b pi\\ / 2/a\\ / 2/b\\
1 + cot |- + --| |1 + cot |- + --||*|1 + cot |- + --|| |1 + tan |-||*|1 + tan |-||
\2 4 / \ \2 4 // \ \2 4 // \ \2// \ \2//
$$\frac{17 \cdot \left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(- \tan^{2}{\left(\frac{b}{2} \right)} + 1\right)}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{b}{2} \right)} + 1\right)} + \frac{17 \cdot \left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(- \cot^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)} + 1\right)}{\left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)} + 1\right)} + \frac{- \cot^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)} + 1}{\cot^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)} + 1}$$
// 0 for a mod pi = 0\ // 0 for b mod pi = 0\ // 0 for b mod pi = 0\ || | || | // 1 for a mod 2*pi = 0\ // 1 for b mod 2*pi = 0\ || | 17*|< / pi\ |*|< / pi\ | + 17*|< |*|< | + |< / pi\ | ||cos|a - --| otherwise | ||cos|b - --| otherwise | \\cos(a) otherwise / \\cos(b) otherwise / ||cos|b - --| otherwise | \\ \ 2 / / \\ \ 2 / / \\ \ 2 / /
$$\left(17 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\cos{\left(b - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(17 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\cos{\left(b \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\cos{\left(b - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\ // 1 for b mod 2*pi = 0\
// 0 for a mod pi = 0\ // 0 for b mod pi = 0\ || | || | // 0 for b mod pi = 0\
|| | || | || 1 | || 1 | || |
17*|< 1 |*|< 1 | + 17*|<----------- otherwise |*|<----------- otherwise | + |< 1 |
||------ otherwise | ||------ otherwise | || /pi \ | || /pi \ | ||------ otherwise |
\\csc(a) / \\csc(b) / ||csc|-- - a| | ||csc|-- - b| | \\csc(b) /
\\ \2 / / \\ \2 / /
$$\left(17 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\csc{\left(a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\frac{1}{\csc{\left(b \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(17 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- b + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\frac{1}{\csc{\left(b \right)}} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\ // 0 for b mod pi = 0\ // 0 for b mod pi = 0\ || | || | // 1 for a mod 2*pi = 0\ // 1 for b mod 2*pi = 0\ || | || 1 | || 1 | || | || | || 1 | 17*|<----------- otherwise |*|<----------- otherwise | + 17*|< 1 |*|< 1 | + |<----------- otherwise | || / pi\ | || / pi\ | ||------ otherwise | ||------ otherwise | || / pi\ | ||sec|a - --| | ||sec|b - --| | \\sec(a) / \\sec(b) / ||sec|b - --| | \\ \ 2 / / \\ \ 2 / / \\ \ 2 / /
$$\left(17 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\frac{1}{\sec{\left(b - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(17 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(b \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\frac{1}{\sec{\left(b - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// / 3*pi\ \ // / 3*pi\ \ // / 3*pi\ \
// 1 for a mod 2*pi = 0\ // 1 for b mod 2*pi = 0\ || 1 for |a + ----| mod 2*pi = 0| || 1 for |b + ----| mod 2*pi = 0| || 1 for |b + ----| mod 2*pi = 0|
17*|< |*|< | + 17*|< \ 2 / |*|< \ 2 / | + |< \ 2 / |
\\cos(a) otherwise / \\cos(b) otherwise / || | || | || |
\\sin(a) otherwise / \\sin(b) otherwise / \\sin(b) otherwise /
$$\left(17 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\cos{\left(b \right)} & \text{otherwise} \end{cases}\right)\right) + \left(17 \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(b + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(b \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \left(b + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(b \right)} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\ // 0 for b mod pi = 0\ // 0 for b mod pi = 0\ || | || | || | ||1 - cos(a) | ||1 - cos(b) | // 1 for a mod 2*pi = 0\ // 1 for b mod 2*pi = 0\ ||1 - cos(b) | 17*|<---------- otherwise |*|<---------- otherwise | + 17*|< |*|< | + |<---------- otherwise | || /a\ | || /b\ | \\cos(a) otherwise / \\cos(b) otherwise / || /b\ | || tan|-| | || tan|-| | || tan|-| | \\ \2/ / \\ \2/ / \\ \2/ /
$$\left(17 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{- \cos{\left(a \right)} + 1}{\tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\frac{- \cos{\left(b \right)} + 1}{\tan{\left(\frac{b}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(17 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\cos{\left(b \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\frac{- \cos{\left(b \right)} + 1}{\tan{\left(\frac{b}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// / pi\ \ // / pi\ \
|| 0 for |a + --| mod pi = 0| || 0 for |b + --| mod pi = 0|
// 0 for a mod pi = 0\ // 0 for b mod pi = 0\ || \ 2 / | || \ 2 / | // 0 for b mod pi = 0\
17*|< |*|< | + 17*|< |*|< | + |< |
\\sin(a) otherwise / \\sin(b) otherwise / || /a pi\ | || /b pi\ | \\sin(b) otherwise /
||(1 + sin(a))*cot|- + --| otherwise | ||(1 + sin(b))*cot|- + --| otherwise |
\\ \2 4 / / \\ \2 4 / /
$$\left(17 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\sin{\left(b \right)} & \text{otherwise} \end{cases}\right)\right) + \left(17 \left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(a \right)} + 1\right) \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(b + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(b \right)} + 1\right) \cot{\left(\frac{b}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\sin{\left(b \right)} & \text{otherwise} \end{cases}\right)$$
/ 4/a\\ / 4/b\\
| 4*sin |-|| | 4*sin |-||
| \2/| | \2/|
2/b\ 17*|1 - ---------|*|1 - ---------| 2/a\ 2/b\
4*sin |-|*sin(b) | 2 | | 2 | 272*sin |-|*sin |-|
\2/ \ sin (a) / \ sin (b) / \2/ \2/
------------------- + ---------------------------------- + ---------------------------------------------
2 4/b\ / 4/a\\ / 4/b\\ / 4/a\\ / 4/b\\
sin (b) + 4*sin |-| | 4*sin |-|| | 4*sin |-|| | 4*sin |-|| | 4*sin |-||
\2/ | \2/| | \2/| | \2/| | \2/|
|1 + ---------|*|1 + ---------| |1 + ---------|*|1 + ---------|*sin(a)*sin(b)
| 2 | | 2 | | 2 | | 2 |
\ sin (a) / \ sin (b) / \ sin (a) / \ sin (b) /
$$\frac{4 \sin^{2}{\left(\frac{b}{2} \right)} \sin{\left(b \right)}}{4 \sin^{4}{\left(\frac{b}{2} \right)} + \sin^{2}{\left(b \right)}} + \frac{17 \left(- \frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right) \left(- \frac{4 \sin^{4}{\left(\frac{b}{2} \right)}}{\sin^{2}{\left(b \right)}} + 1\right)}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right) \left(\frac{4 \sin^{4}{\left(\frac{b}{2} \right)}}{\sin^{2}{\left(b \right)}} + 1\right)} + \frac{272 \sin^{2}{\left(\frac{a}{2} \right)} \sin^{2}{\left(\frac{b}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right) \left(\frac{4 \sin^{4}{\left(\frac{b}{2} \right)}}{\sin^{2}{\left(b \right)}} + 1\right) \sin{\left(a \right)} \sin{\left(b \right)}}$$
/ 4/a\\ / 4/b\\
| 4*sin |-|| | 4*sin |-||
| \2/| | \2/|
2/b\ 17*|1 - ---------|*|1 - ---------| 2/a\ 2/b\
4*sin |-| | 2 | | 2 | 272*sin |-|*sin |-|
\2/ \ sin (a) / \ sin (b) / \2/ \2/
---------------------- + ---------------------------------- + ---------------------------------------------
/ 4/b\\ / 4/a\\ / 4/b\\ / 4/a\\ / 4/b\\
| 4*sin |-|| | 4*sin |-|| | 4*sin |-|| | 4*sin |-|| | 4*sin |-||
| \2/| | \2/| | \2/| | \2/| | \2/|
|1 + ---------|*sin(b) |1 + ---------|*|1 + ---------| |1 + ---------|*|1 + ---------|*sin(a)*sin(b)
| 2 | | 2 | | 2 | | 2 | | 2 |
\ sin (b) / \ sin (a) / \ sin (b) / \ sin (a) / \ sin (b) /
$$\frac{17 \left(- \frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right) \left(- \frac{4 \sin^{4}{\left(\frac{b}{2} \right)}}{\sin^{2}{\left(b \right)}} + 1\right)}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right) \left(\frac{4 \sin^{4}{\left(\frac{b}{2} \right)}}{\sin^{2}{\left(b \right)}} + 1\right)} + \frac{4 \sin^{2}{\left(\frac{b}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{b}{2} \right)}}{\sin^{2}{\left(b \right)}} + 1\right) \sin{\left(b \right)}} + \frac{272 \sin^{2}{\left(\frac{a}{2} \right)} \sin^{2}{\left(\frac{b}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right) \left(\frac{4 \sin^{4}{\left(\frac{b}{2} \right)}}{\sin^{2}{\left(b \right)}} + 1\right) \sin{\left(a \right)} \sin{\left(b \right)}}$$
// 0 for a mod pi = 0\ // 0 for b mod pi = 0\ // 1 for a mod 2*pi = 0\ // 1 for b mod 2*pi = 0\ // 0 for b mod pi = 0\ || | || | || | || | || | || /a\ | || /b\ | || 2/a\ | || 2/b\ | || /b\ | || 2*cot|-| | || 2*cot|-| | ||-1 + cot |-| | ||-1 + cot |-| | || 2*cot|-| | 17*|< \2/ |*|< \2/ | + 17*|< \2/ |*|< \2/ | + |< \2/ | ||----------- otherwise | ||----------- otherwise | ||------------ otherwise | ||------------ otherwise | ||----------- otherwise | || 2/a\ | || 2/b\ | || 2/a\ | || 2/b\ | || 2/b\ | ||1 + cot |-| | ||1 + cot |-| | ||1 + cot |-| | ||1 + cot |-| | ||1 + cot |-| | \\ \2/ / \\ \2/ / \\ \2/ / \\ \2/ / \\ \2/ /
$$\left(17 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{b}{2} \right)}}{\cot^{2}{\left(\frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(17 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{b}{2} \right)} - 1}{\cot^{2}{\left(\frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{b}{2} \right)}}{\cot^{2}{\left(\frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\ // 0 for b mod pi = 0\ // 1 for a mod 2*pi = 0\ // 1 for b mod 2*pi = 0\ // 0 for b mod pi = 0\ || | || | || | || | || | || /a\ | || /b\ | || 2/a\ | || 2/b\ | || /b\ | || 2*tan|-| | || 2*tan|-| | ||1 - tan |-| | ||1 - tan |-| | || 2*tan|-| | 17*|< \2/ |*|< \2/ | + 17*|< \2/ |*|< \2/ | + |< \2/ | ||----------- otherwise | ||----------- otherwise | ||----------- otherwise | ||----------- otherwise | ||----------- otherwise | || 2/a\ | || 2/b\ | || 2/a\ | || 2/b\ | || 2/b\ | ||1 + tan |-| | ||1 + tan |-| | ||1 + tan |-| | ||1 + tan |-| | ||1 + tan |-| | \\ \2/ / \\ \2/ / \\ \2/ / \\ \2/ / \\ \2/ /
$$\left(17 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{b}{2} \right)}}{\tan^{2}{\left(\frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(17 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{a}{2} \right)} + 1}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{b}{2} \right)} + 1}{\tan^{2}{\left(\frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{b}{2} \right)}}{\tan^{2}{\left(\frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\ // 0 for b mod pi = 0\ // 1 for a mod 2*pi = 0\ // 1 for b mod 2*pi = 0\ // 0 for b mod pi = 0\
|| | || | || | || | || |
17*|
$$\left(17 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\sin{\left(b \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(17 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\cos{\left(b \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\sin{\left(b \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
// / pi\ \ // / pi\ \
// 0 for a mod pi = 0\ // 0 for b mod pi = 0\ || 0 for |a + --| mod pi = 0| || 0 for |b + --| mod pi = 0| // 0 for b mod pi = 0\
|| | || | || \ 2 / | || \ 2 / | || |
|| /a\ | || /b\ | || | || | || /b\ |
|| 2*cot|-| | || 2*cot|-| | || /a pi\ | || /b pi\ | || 2*cot|-| |
17*|< \2/ |*|< \2/ | + 17*|< 2*cot|- + --| |*|< 2*cot|- + --| | + |< \2/ |
||----------- otherwise | ||----------- otherwise | || \2 4 / | || \2 4 / | ||----------- otherwise |
|| 2/a\ | || 2/b\ | ||---------------- otherwise | ||---------------- otherwise | || 2/b\ |
||1 + cot |-| | ||1 + cot |-| | || 2/a pi\ | || 2/b pi\ | ||1 + cot |-| |
\\ \2/ / \\ \2/ / ||1 + cot |- + --| | ||1 + cot |- + --| | \\ \2/ /
\\ \2 4 / / \\ \2 4 / /
$$\left(17 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{b}{2} \right)}}{\cot^{2}{\left(\frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(17 \left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(b + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{b}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{b}{2} \right)}}{\cot^{2}{\left(\frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\ // 1 for b mod 2*pi = 0\
|| | || |
// 0 for a mod pi = 0\ // 0 for b mod pi = 0\ || 1 | || 1 | // 0 for b mod pi = 0\
|| | || | ||-1 + ------- | ||-1 + ------- | || |
|| 2 | || 2 | || 2/a\ | || 2/b\ | || 2 |
||-------------------- otherwise | ||-------------------- otherwise | || tan |-| | || tan |-| | ||-------------------- otherwise |
17*|
$$\left(17 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right) \tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{b}{2} \right)}}\right) \tan{\left(\frac{b}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(17 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{b}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{b}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{b}{2} \right)}}\right) \tan{\left(\frac{b}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
/ 2/a\ \ / 2/b\ \
| sec |-| | | sec |-| |
| \2/ | | \2/ |
17*|1 - ------------|*|1 - ------------|
/b\ | 2/a pi\| | 2/b pi\| /a\ /b\
2*sec|-| | sec |- - --|| | sec |- - --|| 68*sec|-|*sec|-|
\2/ \ \2 2 // \ \2 2 // \2/ \2/
------------------------------ + ---------------------------------------- + -------------------------------------------------------------
/ 2/b\ \ / 2/a\ \ / 2/b\ \ / 2/a\ \ / 2/b\ \
| sec |-| | | sec |-| | | sec |-| | | sec |-| | | sec |-| |
| \2/ | /b pi\ | \2/ | | \2/ | | \2/ | | \2/ | /a pi\ /b pi\
|1 + ------------|*sec|- - --| |1 + ------------|*|1 + ------------| |1 + ------------|*|1 + ------------|*sec|- - --|*sec|- - --|
| 2/b pi\| \2 2 / | 2/a pi\| | 2/b pi\| | 2/a pi\| | 2/b pi\| \2 2 / \2 2 /
| sec |- - --|| | sec |- - --|| | sec |- - --|| | sec |- - --|| | sec |- - --||
\ \2 2 // \ \2 2 // \ \2 2 // \ \2 2 // \ \2 2 //
$$\frac{17 \left(- \frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \left(- \frac{\sec^{2}{\left(\frac{b}{2} \right)}}{\sec^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}} + 1\right)}{\left(\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \left(\frac{\sec^{2}{\left(\frac{b}{2} \right)}}{\sec^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}} + 1\right)} + \frac{2 \sec{\left(\frac{b}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{b}{2} \right)}}{\sec^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{b}{2} - \frac{\pi}{2} \right)}} + \frac{68 \sec{\left(\frac{a}{2} \right)} \sec{\left(\frac{b}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \left(\frac{\sec^{2}{\left(\frac{b}{2} \right)}}{\sec^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)} \sec{\left(\frac{b}{2} - \frac{\pi}{2} \right)}}$$
/ 2/a pi\\ / 2/b pi\\
| cos |- - --|| | cos |- - --||
| \2 2 /| | \2 2 /|
17*|1 - ------------|*|1 - ------------|
/b pi\ | 2/a\ | | 2/b\ | /a pi\ /b pi\
2*cos|- - --| | cos |-| | | cos |-| | 68*cos|- - --|*cos|- - --|
\2 2 / \ \2/ / \ \2/ / \2 2 / \2 2 /
------------------------- + ---------------------------------------- + ---------------------------------------------------
/ 2/b pi\\ / 2/a pi\\ / 2/b pi\\ / 2/a pi\\ / 2/b pi\\
| cos |- - --|| | cos |- - --|| | cos |- - --|| | cos |- - --|| | cos |- - --||
| \2 2 /| /b\ | \2 2 /| | \2 2 /| | \2 2 /| | \2 2 /| /a\ /b\
|1 + ------------|*cos|-| |1 + ------------|*|1 + ------------| |1 + ------------|*|1 + ------------|*cos|-|*cos|-|
| 2/b\ | \2/ | 2/a\ | | 2/b\ | | 2/a\ | | 2/b\ | \2/ \2/
| cos |-| | | cos |-| | | cos |-| | | cos |-| | | cos |-| |
\ \2/ / \ \2/ / \ \2/ / \ \2/ / \ \2/ /
$$\frac{17 \cdot \left(1 - \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right) \left(1 - \frac{\cos^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{b}{2} \right)}}\right)}{\left(1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right) \left(1 + \frac{\cos^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{b}{2} \right)}}\right)} + \frac{2 \cos{\left(\frac{b}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{b}{2} \right)}}\right) \cos{\left(\frac{b}{2} \right)}} + \frac{68 \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)} \cos{\left(\frac{b}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right) \left(1 + \frac{\cos^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{b}{2} \right)}}\right) \cos{\left(\frac{a}{2} \right)} \cos{\left(\frac{b}{2} \right)}}$$
/ 2/pi a\\ / 2/pi b\\
| csc |-- - -|| | csc |-- - -||
| \2 2/| | \2 2/|
17*|1 - ------------|*|1 - ------------|
/pi b\ | 2/a\ | | 2/b\ | /pi a\ /pi b\
2*csc|-- - -| | csc |-| | | csc |-| | 68*csc|-- - -|*csc|-- - -|
\2 2/ \ \2/ / \ \2/ / \2 2/ \2 2/
------------------------- + ---------------------------------------- + ---------------------------------------------------
/ 2/pi b\\ / 2/pi a\\ / 2/pi b\\ / 2/pi a\\ / 2/pi b\\
| csc |-- - -|| | csc |-- - -|| | csc |-- - -|| | csc |-- - -|| | csc |-- - -||
| \2 2/| /b\ | \2 2/| | \2 2/| | \2 2/| | \2 2/| /a\ /b\
|1 + ------------|*csc|-| |1 + ------------|*|1 + ------------| |1 + ------------|*|1 + ------------|*csc|-|*csc|-|
| 2/b\ | \2/ | 2/a\ | | 2/b\ | | 2/a\ | | 2/b\ | \2/ \2/
| csc |-| | | csc |-| | | csc |-| | | csc |-| | | csc |-| |
\ \2/ / \ \2/ / \ \2/ / \ \2/ / \ \2/ /
$$\frac{17 \cdot \left(1 - \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right) \left(1 - \frac{\csc^{2}{\left(- \frac{b}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{b}{2} \right)}}\right)}{\left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right) \left(1 + \frac{\csc^{2}{\left(- \frac{b}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{b}{2} \right)}}\right)} + \frac{2 \csc{\left(- \frac{b}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{b}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{b}{2} \right)}}\right) \csc{\left(\frac{b}{2} \right)}} + \frac{68 \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)} \csc{\left(- \frac{b}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right) \left(1 + \frac{\csc^{2}{\left(- \frac{b}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{b}{2} \right)}}\right) \csc{\left(\frac{a}{2} \right)} \csc{\left(\frac{b}{2} \right)}}$$
// / 3*pi\ \ // / 3*pi\ \ // / 3*pi\ \
// 1 for a mod 2*pi = 0\ // 1 for b mod 2*pi = 0\ || 1 for |a + ----| mod 2*pi = 0| || 1 for |b + ----| mod 2*pi = 0| || 1 for |b + ----| mod 2*pi = 0|
|| | || | || \ 2 / | || \ 2 / | || \ 2 / |
|| 2/a\ | || 2/b\ | || | || | || |
||-1 + cot |-| | ||-1 + cot |-| | || 2/a pi\ | || 2/b pi\ | || 2/b pi\ |
17*|< \2/ |*|< \2/ | + 17*|<-1 + tan |- + --| |*|<-1 + tan |- + --| | + |<-1 + tan |- + --| |
||------------ otherwise | ||------------ otherwise | || \2 4 / | || \2 4 / | || \2 4 / |
|| 2/a\ | || 2/b\ | ||----------------- otherwise | ||----------------- otherwise | ||----------------- otherwise |
||1 + cot |-| | ||1 + cot |-| | || 2/a pi\ | || 2/b pi\ | || 2/b pi\ |
\\ \2/ / \\ \2/ / || 1 + tan |- + --| | || 1 + tan |- + --| | || 1 + tan |- + --| |
\\ \2 4 / / \\ \2 4 / / \\ \2 4 / /
$$\left(17 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{b}{2} \right)} - 1}{\cot^{2}{\left(\frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(17 \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(b + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \left(b + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\ // 0 for b mod pi = 0\ // 0 for b mod pi = 0\ || | || | || | || 2*sin(a) | || 2*sin(b) | // 1 for a mod 2*pi = 0\ // 1 for b mod 2*pi = 0\ || 2*sin(b) | ||---------------------------- otherwise | ||---------------------------- otherwise | || | || | ||---------------------------- otherwise | || / 2 \ | || / 2 \ | || 2 | || 2 | || / 2 \ | 17*|< | sin (a) | |*|< | sin (b) | | + 17*|< -4 + 4*sin (a) + 4*cos(a) |*|< -4 + 4*sin (b) + 4*cos(b) | + |< | sin (b) | | ||(1 - cos(a))*|1 + ---------| | ||(1 - cos(b))*|1 + ---------| | ||--------------------------- otherwise | ||--------------------------- otherwise | ||(1 - cos(b))*|1 + ---------| | || | 4/a\| | || | 4/b\| | || 2 2 | || 2 2 | || | 4/b\| | || | 4*sin |-|| | || | 4*sin |-|| | \\2*(1 - cos(a)) + 2*sin (a) / \\2*(1 - cos(b)) + 2*sin (b) / || | 4*sin |-|| | || \ \2// | || \ \2// | || \ \2// | \\ / \\ / \\ /
$$\left(17 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \sin{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \left(- \cos{\left(a \right)} + 1\right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\frac{2 \sin{\left(b \right)}}{\left(1 + \frac{\sin^{2}{\left(b \right)}}{4 \sin^{4}{\left(\frac{b}{2} \right)}}\right) \left(- \cos{\left(b \right)} + 1\right)} & \text{otherwise} \end{cases}\right)\right) + \left(17 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{4 \sin^{2}{\left(a \right)} + 4 \cos{\left(a \right)} - 4}{2 \left(- \cos{\left(a \right)} + 1\right)^{2} + 2 \sin^{2}{\left(a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{4 \sin^{2}{\left(b \right)} + 4 \cos{\left(b \right)} - 4}{2 \left(- \cos{\left(b \right)} + 1\right)^{2} + 2 \sin^{2}{\left(b \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\frac{2 \sin{\left(b \right)}}{\left(1 + \frac{\sin^{2}{\left(b \right)}}{4 \sin^{4}{\left(\frac{b}{2} \right)}}\right) \left(- \cos{\left(b \right)} + 1\right)} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\ // 1 for b mod 2*pi = 0\
|| | || |
// 0 for a mod pi = 0\ // 0 for b mod pi = 0\ || 2 | || 2 | // 0 for b mod pi = 0\
|| | || | || sin (a) | || sin (b) | || |
|| sin(a) | || sin(b) | ||-1 + --------- | ||-1 + --------- | || sin(b) |
||----------------------- otherwise | ||----------------------- otherwise | || 4/a\ | || 4/b\ | ||----------------------- otherwise |
||/ 2 \ | ||/ 2 \ | || 4*sin |-| | || 4*sin |-| | ||/ 2 \ |
17*|<| sin (a) | 2/a\ |*|<| sin (b) | 2/b\ | + 17*|< \2/ |*|< \2/ | + |<| sin (b) | 2/b\ |
|||1 + ---------|*sin |-| | |||1 + ---------|*sin |-| | ||-------------- otherwise | ||-------------- otherwise | |||1 + ---------|*sin |-| |
||| 4/a\| \2/ | ||| 4/b\| \2/ | || 2 | || 2 | ||| 4/b\| \2/ |
||| 4*sin |-|| | ||| 4*sin |-|| | || sin (a) | || sin (b) | ||| 4*sin |-|| |
||\ \2// | ||\ \2// | ||1 + --------- | ||1 + --------- | ||\ \2// |
\\ / \\ / || 4/a\ | || 4/b\ | \\ /
|| 4*sin |-| | || 4*sin |-| |
\\ \2/ / \\ \2/ /
$$\left(17 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{\sin{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \sin^{2}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\frac{\sin{\left(b \right)}}{\left(1 + \frac{\sin^{2}{\left(b \right)}}{4 \sin^{4}{\left(\frac{b}{2} \right)}}\right) \sin^{2}{\left(\frac{b}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(17 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}}{1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(b \right)}}{4 \sin^{4}{\left(\frac{b}{2} \right)}}}{1 + \frac{\sin^{2}{\left(b \right)}}{4 \sin^{4}{\left(\frac{b}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\frac{\sin{\left(b \right)}}{\left(1 + \frac{\sin^{2}{\left(b \right)}}{4 \sin^{4}{\left(\frac{b}{2} \right)}}\right) \sin^{2}{\left(\frac{b}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\ // 0 for b mod pi = 0\ // 1 for a mod 2*pi = 0\ // 1 for b mod 2*pi = 0\ // 0 for b mod pi = 0\ || | || | || | || | || | ||/ 0 for a mod pi = 0 | ||/ 0 for b mod pi = 0 | ||/ 1 for a mod 2*pi = 0 | ||/ 1 for b mod 2*pi = 0 | ||/ 0 for b mod pi = 0 | ||| | ||| | ||| | ||| | ||| | ||| /a\ | ||| /b\ | ||| 2/a\ | ||| 2/b\ | ||| /b\ | 17*|<| 2*cot|-| |*|<| 2*cot|-| | + 17*|<|-1 + cot |-| |*|<|-1 + cot |-| | + |<| 2*cot|-| | ||< \2/ otherwise | ||< \2/ otherwise | ||< \2/ otherwise | ||< \2/ otherwise | ||< \2/ otherwise | |||----------- otherwise | |||----------- otherwise | |||------------ otherwise | |||------------ otherwise | |||----------- otherwise | ||| 2/a\ | ||| 2/b\ | ||| 2/a\ | ||| 2/b\ | ||| 2/b\ | |||1 + cot |-| | |||1 + cot |-| | |||1 + cot |-| | |||1 + cot |-| | |||1 + cot |-| | \\\ \2/ / \\\ \2/ / \\\ \2/ / \\\ \2/ / \\\ \2/ /
$$\left(17 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{b}{2} \right)}}{\cot^{2}{\left(\frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(17 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{b}{2} \right)} - 1}{\cot^{2}{\left(\frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{b}{2} \right)}}{\cot^{2}{\left(\frac{b}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\ // 1 for b mod 2*pi = 0\
|| | || |
// 0 for a mod pi = 0\ // 0 for b mod pi = 0\ || 2/a\ | || 2/b\ | // 0 for b mod pi = 0\
|| | || | || cos |-| | || cos |-| | || |
|| /a\ | || /b\ | || \2/ | || \2/ | || /b\ |
|| 2*cos|-| | || 2*cos|-| | ||-1 + ------------ | ||-1 + ------------ | || 2*cos|-| |
|| \2/ | || \2/ | || 2/a pi\ | || 2/b pi\ | || \2/ |
||------------------------------ otherwise | ||------------------------------ otherwise | || cos |- - --| | || cos |- - --| | ||------------------------------ otherwise |
17*|
$$\left(17 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{a}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{b}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{b}{2} \right)}}{\cos^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{b}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(17 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{b}{2} \right)}}{\cos^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{b}{2} \right)}}{\cos^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{b}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{b}{2} \right)}}{\cos^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{b}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\ // 1 for b mod 2*pi = 0\
|| | || |
// 0 for a mod pi = 0\ // 0 for b mod pi = 0\ || 2/a pi\ | || 2/b pi\ | // 0 for b mod pi = 0\
|| | || | || sec |- - --| | || sec |- - --| | || |
|| /a pi\ | || /b pi\ | || \2 2 / | || \2 2 / | || /b pi\ |
|| 2*sec|- - --| | || 2*sec|- - --| | ||-1 + ------------ | ||-1 + ------------ | || 2*sec|- - --| |
|| \2 2 / | || \2 2 / | || 2/a\ | || 2/b\ | || \2 2 / |
||------------------------- otherwise | ||------------------------- otherwise | || sec |-| | || sec |-| | ||------------------------- otherwise |
17*|
$$\left(17 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right) \sec{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{b}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{b}{2} \right)}}\right) \sec{\left(\frac{b}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(17 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{b}{2} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{b}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{b}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{b}{2} \right)}}\right) \sec{\left(\frac{b}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\ // 1 for b mod 2*pi = 0\
|| | || |
// 0 for a mod pi = 0\ // 0 for b mod pi = 0\ || 2/a\ | || 2/b\ | // 0 for b mod pi = 0\
|| | || | || csc |-| | || csc |-| | || |
|| /a\ | || /b\ | || \2/ | || \2/ | || /b\ |
|| 2*csc|-| | || 2*csc|-| | ||-1 + ------------ | ||-1 + ------------ | || 2*csc|-| |
|| \2/ | || \2/ | || 2/pi a\ | || 2/pi b\ | || \2/ |
||------------------------------ otherwise | ||------------------------------ otherwise | || csc |-- - -| | || csc |-- - -| | ||------------------------------ otherwise |
17*|
$$\left(17 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{a}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{b}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{b}{2} \right)}}{\csc^{2}{\left(- \frac{b}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{b}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(17 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{b}{2} \right)}}{\csc^{2}{\left(- \frac{b}{2} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{b}{2} \right)}}{\csc^{2}{\left(- \frac{b}{2} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{b}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{b}{2} \right)}}{\csc^{2}{\left(- \frac{b}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{b}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
17*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((0, Mod(b = pi, 0)), (2*csc(b/2)/((1 + csc(b/2)^2/csc(pi/2 - b/2)^2)*csc(pi/2 - b/2)), True)) + 17*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((0, Mod(b = pi, 0)), (2*csc(b/2)/((1 + csc(b/2)^2/csc(pi/2 - b/2)^2)*csc(pi/2 - b/2)), True))
Объединение рациональных выражений
[src]
17*cos(a)*cos(b) + 17*sin(a)*sin(b) + sin(b)
$$17 \sin{\left(a \right)} \sin{\left(b \right)} + 17 \cos{\left(a \right)} \cos{\left(b \right)} + \sin{\left(b \right)}$$
17*cos(a)*cos(b) + 17*sin(a)*sin(b) + sin(b)