Господин Экзамен
Общий знаменатель sin(pi+a)/sin((pi/2)-a)
Решение
Вы ввели
[src]
sin(pi + a) ----------- /pi \ sin|-- - a| \2 /
$$\frac{\sin{\left(a + \pi \right)}}{\sin{\left(- a + \frac{\pi}{2} \right)}}$$
sin(pi + a)/sin(pi/2 - a)
Рациональный знаменатель
[src]
-sin(a) -------- cos(a)
$$- \frac{\sin{\left(a \right)}}{\cos{\left(a \right)}}$$
-sin(a)/cos(a)
Объединение рациональных выражений
[src]
-sin(a) ------------- /pi - 2*a\ sin|--------| \ 2 /
$$- \frac{\sin{\left(a \right)}}{\sin{\left(\frac{- 2 a + \pi}{2} \right)}}$$
-sin(a)/sin((pi - 2*a)/2)
Степени
[src]
-sin(a) -------- cos(a)
$$- \frac{\sin{\left(a \right)}}{\cos{\left(a \right)}}$$
I*(-pi - a) I*(pi + a)
- e + e
----------------------------
/ pi\ /pi \
I*|a - --| I*|-- - a|
\ 2 / \2 /
- e + e
$$\frac{- e^{i \left(- a - \pi\right)} + e^{i \left(a + \pi\right)}}{e^{i \left(- a + \frac{\pi}{2}\right)} - e^{i \left(a - \frac{\pi}{2}\right)}}$$
(-exp(i*(-pi - a)) + exp(i*(pi + a)))/(-exp(i*(a - pi/2)) + exp(i*(pi/2 - a)))
Комбинаторика
[src]
-sin(a) -------- cos(a)
$$- \frac{\sin{\left(a \right)}}{\cos{\left(a \right)}}$$
-sin(a)/cos(a)
Общий знаменатель
[src]
sin(pi + a) ----------- cos(a)
$$\frac{\sin{\left(a + \pi \right)}}{\cos{\left(a \right)}}$$
sin(pi + a)/cos(a)
Собрать выражение
[src]
-sec(a)*sin(a)
$$- \sin{\left(a \right)} \sec{\left(a \right)}$$
-sec(a)*sin(a)
Тригонометрическая часть
[src]
-tan(a)
$$- \tan{\left(a \right)}$$
-1 ------ cot(a)
$$- \frac{1}{\cot{\left(a \right)}}$$
-sec(a) -------- csc(a)
$$- \frac{\sec{\left(a \right)}}{\csc{\left(a \right)}}$$
-sin(a) -------- cos(a)
$$- \frac{\sin{\left(a \right)}}{\cos{\left(a \right)}}$$
2 -2*sin (a) ---------- sin(2*a)
$$- \frac{2 \sin^{2}{\left(a \right)}}{\sin{\left(2 a \right)}}$$
-sin(a) ----------- / pi\ sin|a + --| \ 2 /
$$- \frac{\sin{\left(a \right)}}{\sin{\left(a + \frac{\pi}{2} \right)}}$$
-sec(a) ----------- / pi\ sec|a - --| \ 2 /
$$- \frac{\sec{\left(a \right)}}{\sec{\left(a - \frac{\pi}{2} \right)}}$$
/ pi\
-cos|a - --|
\ 2 /
-------------
cos(a)
$$- \frac{\cos{\left(a - \frac{\pi}{2} \right)}}{\cos{\left(a \right)}}$$
/pi \
-csc|-- - a|
\2 /
-------------
csc(a)
$$- \frac{\csc{\left(- a + \frac{\pi}{2} \right)}}{\csc{\left(a \right)}}$$
-sec(a) ----------- /pi \ sec|-- - a| \2 /
$$- \frac{\sec{\left(a \right)}}{\sec{\left(- a + \frac{\pi}{2} \right)}}$$
/pi \
-csc|-- - a|
\2 /
-------------
csc(pi - a)
$$- \frac{\csc{\left(- a + \frac{\pi}{2} \right)}}{\csc{\left(- a + \pi \right)}}$$
/ 1 \ /a\ -|1 + ------|*tan|-| \ cos(a)/ \2/
$$- \left(1 + \frac{1}{\cos{\left(a \right)}}\right) \tan{\left(\frac{a}{2} \right)}$$
/a\
-2*tan|-|
\2/
-----------
2/a\
1 - tan |-|
\2/
$$- \frac{2 \tan{\left(\frac{a}{2} \right)}}{- \tan^{2}{\left(\frac{a}{2} \right)} + 1}$$
-2 -------------------- / 1 \ /a\ |1 - -------|*cot|-| | 2/a\| \2/ | cot |-|| \ \2//
$$- \frac{2}{\left(1 - \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \cot{\left(\frac{a}{2} \right)}}$$
sin(a) ---------------------- / 1 \ 2/a\ |-2 + -------|*cos |-| | 2/a\| \2/ | cos |-|| \ \2//
$$\frac{\sin{\left(a \right)}}{\left(-2 + \frac{1}{\cos^{2}{\left(\frac{a}{2} \right)}}\right) \cos^{2}{\left(\frac{a}{2} \right)}}$$
2/a\
-4*sin |-|*sin(a)
\2/
-------------------
2 4/a\
sin (a) - 4*sin |-|
\2/
$$- \frac{4 \sin^{2}{\left(\frac{a}{2} \right)} \sin{\left(a \right)}}{- 4 \sin^{4}{\left(\frac{a}{2} \right)} + \sin^{2}{\left(a \right)}}$$
2/a\
-4*sin |-|
\2/
----------------------
/ 4/a\\
| 4*sin |-||
| \2/|
|1 - ---------|*sin(a)
| 2 |
\ sin (a) /
$$- \frac{4 \sin^{2}{\left(\frac{a}{2} \right)}}{\left(- \frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right) \sin{\left(a \right)}}$$
/a pi\
-2*cos|- - --|
\2 2 /
-------------------------
/ 2/a pi\\
| cos |- - --||
| \2 2 /| /a\
|1 - ------------|*cos|-|
| 2/a\ | \2/
| cos |-| |
\ \2/ /
$$- \frac{2 \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 - \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right) \cos{\left(\frac{a}{2} \right)}}$$
/a\
-2*sec|-|
\2/
------------------------------
/ 2/a\ \
| sec |-| |
| \2/ | /a pi\
|1 - ------------|*sec|- - --|
| 2/a pi\| \2 2 /
| sec |- - --||
\ \2 2 //
$$- \frac{2 \sec{\left(\frac{a}{2} \right)}}{\left(- \frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}$$
/pi a\
-2*csc|-- - -|
\2 2/
-------------------------
/ 2/pi a\\
| csc |-- - -||
| \2 2/| /a\
|1 - ------------|*csc|-|
| 2/a\ | \2/
| csc |-| |
\ \2/ /
$$- \frac{2 \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\left(1 - \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right) \csc{\left(\frac{a}{2} \right)}}$$
/ 2/a pi\\ /a\ -|1 + tan |- + --||*cot|-| \ \2 4 // \2/ --------------------------- / 2/a\\ /a pi\ |1 + cot |-||*tan|- + --| \ \2// \2 4 /
$$- \frac{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \cot{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right) \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
/ 2/a pi\\ /a\ -|1 + tan |- + --||*tan|-| \ \2 4 // \2/ --------------------------- / 2/a\\ /a pi\ |1 + tan |-||*tan|- + --| \ \2// \2 4 /
$$- \frac{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \tan{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
/ 2/a pi\\
-|1 - cot |- + --||*(1 + sin(a))
\ \2 4 //
---------------------------------
/ 2/a\\ 2/a\
2*|1 - tan |-||*cos |-|
\ \2// \2/
$$- \frac{\left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(a \right)} + 1\right)}{2 \cdot \left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \cos^{2}{\left(\frac{a}{2} \right)}}$$
// 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\ -|< |*|< | \\sin(a) otherwise / \\sec(a) otherwise /
$$- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\sec{\left(a \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\
// 0 for a mod pi = 0\ || |
-|< |*|< 1 |
\\sin(a) otherwise / ||------ otherwise |
\\cos(a) /
$$- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\cos{\left(a \right)}} & \text{otherwise} \end{cases}\right)$$
/ 2/a\\ / 2/a pi\\ -|1 + cot |-||*|-1 + tan |- + --|| \ \2// \ \2 4 // ----------------------------------- / 2/a pi\\ / 2/a\\ |1 + tan |- + --||*|-1 + cot |-|| \ \2 4 // \ \2//
$$- \frac{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)}$$
/ 2/a\\ / 2/a pi\\ -|1 + tan |-||*|1 - cot |- + --|| \ \2// \ \2 4 // ---------------------------------- / 2/a pi\\ / 2/a\\ |1 + cot |- + --||*|1 - tan |-|| \ \2 4 // \ \2//
$$- \frac{\left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}{\left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)}$$
// 1 for a mod 2*pi = 0\
|| |
// 0 for a mod pi = 0\ || 1 |
-|< |*|<----------- otherwise |
\\sin(a) otherwise / || / pi\ |
||sin|a + --| |
\\ \ 2 / /
$$- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\sin{\left(a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\ || | || 1 | // 1 for a mod 2*pi = 0\ -|<----------- otherwise |*|< | || / pi\ | \\sec(a) otherwise / ||sec|a - --| | \\ \ 2 / /
$$- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\sec{\left(a \right)} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\ || | || | -|< / pi\ |*|< 1 | ||cos|a - --| otherwise | ||------ otherwise | \\ \ 2 / / \\cos(a) /
$$- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\cos{\left(a \right)}} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\ || | || | -|< 1 |*|< /pi \ | ||------ otherwise | ||csc|-- - a| otherwise | \\csc(a) / \\ \2 / /
$$- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\csc{\left(a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\csc{\left(- a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\ // / 3*pi\ \ || | || 1 for |a + ----| mod 2*pi = 0| -|< 1 |*|< \ 2 / | ||------ otherwise | || | \\cos(a) / \\sin(a) otherwise /
$$- \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\cos{\left(a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\ || | // 1 for a mod 2*pi = 0\ ||1 - cos(a) | || | -|<---------- otherwise |*|< 1 | || /a\ | ||------ otherwise | || tan|-| | \\cos(a) / \\ \2/ /
$$- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{- \cos{\left(a \right)} + 1}{\tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\cos{\left(a \right)}} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\ || | || | || /a\ | || 2/a\ | || 2*cot|-| | ||1 + cot |-| | -|< \2/ |*|< \2/ | ||----------- otherwise | ||------------ otherwise | || 2/a\ | || 2/a\ | ||1 + cot |-| | ||-1 + cot |-| | \\ \2/ / \\ \2/ /
$$- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} + 1}{\cot^{2}{\left(\frac{a}{2} \right)} - 1} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\ || | || | || /a\ | || 2/a\ | || 2*tan|-| | ||1 + tan |-| | -|< \2/ |*|< \2/ | ||----------- otherwise | ||----------- otherwise | || 2/a\ | || 2/a\ | ||1 + tan |-| | ||1 - tan |-| | \\ \2/ / \\ \2/ /
$$- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{a}{2} \right)} + 1}{- \tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// / pi\ \
|| zoo for |a + --| mod pi = 0|
|| \ 2 / |
|| |
// 0 for a mod pi = 0\ || /a pi\ |
-|< |*|< tan|- + --| |
\\sin(a) otherwise / || \2 4 / |
||-------------- otherwise |
|| 2/a pi\ |
||2*sin |- + --| |
\\ \2 4 / /
$$- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{\tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{2 \sin^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\
|| |
// 0 for a mod pi = 0\ || 1 |
|| | ||1 + ------- |
|| 2 | || 2/a\ |
||-------------------- otherwise | || tan |-| |
-|
$$- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right) \tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}}{-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\
// 0 for a mod pi = 0\ || |
|| | ||/ 1 for a mod 2*pi = 0 |
-|
$$- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\cos{\left(a \right)}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
// / pi\ \
// 0 for a mod pi = 0\ || zoo for |a + --| mod pi = 0|
|| | || \ 2 / |
|| /a\ | || |
|| 2*cot|-| | || 2/a pi\ |
-|< \2/ |*|<1 + cot |- + --| |
||----------- otherwise | || \2 4 / |
|| 2/a\ | ||---------------- otherwise |
||1 + cot |-| | || /a pi\ |
\\ \2/ / || 2*cot|- + --| |
\\ \2 4 / /
$$- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1}{2 \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)}} & \text{otherwise} \end{cases}\right)$$
// / 3*pi\ \
// 1 for a mod 2*pi = 0\ || 1 for |a + ----| mod 2*pi = 0|
|| | || \ 2 / |
|| 2/a\ | || |
||1 + cot |-| | || 2/a pi\ |
-|< \2/ |*|<-1 + tan |- + --| |
||------------ otherwise | || \2 4 / |
|| 2/a\ | ||----------------- otherwise |
||-1 + cot |-| | || 2/a pi\ |
\\ \2/ / || 1 + tan |- + --| |
\\ \2 4 / /
$$- \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} + 1}{\cot^{2}{\left(\frac{a}{2} \right)} - 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\ || | || 2*sin(a) | // 1 for a mod 2*pi = 0\ ||---------------------------- otherwise | || | || / 2 \ | || 2 2 | -|< | sin (a) | |*|< (1 - cos(a)) + sin (a) | ||(1 - cos(a))*|1 + ---------| | ||------------------------- otherwise | || | 4/a\| | || 2 | || | 4*sin |-|| | \\-2 + 2*sin (a) + 2*cos(a) / || \ \2// | \\ /
$$- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \sin{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \left(- \cos{\left(a \right)} + 1\right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(- \cos{\left(a \right)} + 1\right)^{2} + \sin^{2}{\left(a \right)}}{2 \sin^{2}{\left(a \right)} + 2 \cos{\left(a \right)} - 2} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\
|| |
// 0 for a mod pi = 0\ || 2 |
|| | || sin (a) |
|| sin(a) | ||1 + --------- |
||----------------------- otherwise | || 4/a\ |
||/ 2 \ | || 4*sin |-| |
-|<| sin (a) | 2/a\ |*|< \2/ |
|||1 + ---------|*sin |-| | ||-------------- otherwise |
||| 4/a\| \2/ | || 2 |
||| 4*sin |-|| | || sin (a) |
||\ \2// | ||-1 + --------- |
\\ / || 4/a\ |
|| 4*sin |-| |
\\ \2/ /
$$- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{\sin{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \sin^{2}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}}{-1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\ || | || | ||/ 0 for a mod pi = 0 | ||/ 1 for a mod 2*pi = 0 | ||| | ||| | ||| /a\ | ||| 2/a\ | -|<| 2*cot|-| |*|<|1 + cot |-| | ||< \2/ otherwise | ||< \2/ otherwise | |||----------- otherwise | |||------------ otherwise | ||| 2/a\ | ||| 2/a\ | |||1 + cot |-| | |||-1 + cot |-| | \\\ \2/ / \\\ \2/ /
$$- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} + 1}{\cot^{2}{\left(\frac{a}{2} \right)} - 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\
|| |
// 0 for a mod pi = 0\ || 2/a\ |
|| | || cos |-| |
|| /a\ | || \2/ |
|| 2*cos|-| | || 1 + ------------ |
|| \2/ | || 2/a pi\ |
||------------------------------ otherwise | || cos |- - --| |
-|
$$- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{a}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1}{\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - 1} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\
|| |
// 0 for a mod pi = 0\ || 2/a pi\ |
|| | || sec |- - --| |
|| /a pi\ | || \2 2 / |
|| 2*sec|- - --| | || 1 + ------------ |
|| \2 2 / | || 2/a\ |
||------------------------- otherwise | || sec |-| |
-|
$$- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right) \sec{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}}{-1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\
|| |
// 0 for a mod pi = 0\ || 2/a\ |
|| | || csc |-| |
|| /a\ | || \2/ |
|| 2*csc|-| | || 1 + ------------ |
|| \2/ | || 2/pi a\ |
||------------------------------ otherwise | || csc |-- - -| |
-|
$$- \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{a}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1}{\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} - 1} & \text{otherwise} \end{cases}\right)$$
-Piecewise((0, Mod(a = pi, 0)), (2*csc(a/2)/((1 + csc(a/2)^2/csc(pi/2 - a/2)^2)*csc(pi/2 - a/2)), True))*Piecewise((1, Mod(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))