Господин Экзамен
Общий знаменатель sin(pi+a)*cos(pi-a)/tan((3*pi/2)-a)
Решение
Вы ввели
[src]
sin(pi + a)*cos(pi - a)
-----------------------
/3*pi \
tan|---- - a|
\ 2 /
$$\frac{\sin{\left(a + \pi \right)} \cos{\left(- a + \pi \right)}}{\tan{\left(- a + \frac{3 \pi}{2} \right)}}$$
sin(pi + a)*cos(pi - a)/tan(3*pi/2 - a)
Численный ответ
[src]
cos(pi - a)*sin(pi + a)/tan(3*pi/2 - a)
cos(pi - a)*sin(pi + a)/tan(3*pi/2 - a)
Рациональный знаменатель
[src]
cos(a)*sin(a)
-------------
cot(a)
$$\frac{\sin{\left(a \right)} \cos{\left(a \right)}}{\cot{\left(a \right)}}$$
cos(a)*sin(a)/cot(a)
Комбинаторика
[src]
cos(a)*sin(a)
-------------
cot(a)
$$\frac{\sin{\left(a \right)} \cos{\left(a \right)}}{\cot{\left(a \right)}}$$
cos(a)*sin(a)/cot(a)
Общий знаменатель
[src]
-cos(a)*sin(pi + a)
--------------------
cot(a)
$$- \frac{\sin{\left(a + \pi \right)} \cos{\left(a \right)}}{\cot{\left(a \right)}}$$
-cos(a)*sin(pi + a)/cot(a)
Объединение рациональных выражений
[src]
cos(a)*sin(a) ---------------- /-2*a + 3*pi\ tan|-----------| \ 2 /
$$\frac{\sin{\left(a \right)} \cos{\left(a \right)}}{\tan{\left(\frac{- 2 a + 3 \pi}{2} \right)}}$$
cos(a)*sin(a)/tan((-2*a + 3*pi)/2)
Собрать выражение
[src]
sin(2*a)*tan(a)
---------------
2
$$\frac{\sin{\left(2 a \right)} \tan{\left(a \right)}}{2}$$
sin(2*a)*tan(a)/2
Степени
[src]
cos(a)*sin(a)
-------------
cot(a)
$$\frac{\sin{\left(a \right)} \cos{\left(a \right)}}{\cot{\left(a \right)}}$$
/ / 3*pi\ / 3*pi\\
/ I*(pi - a) I*(a - pi)\ | I*|a - ----| I*|-a + ----||
|e e | / I*(-pi - a) I*(pi + a)\ | \ 2 / \ 2 /|
-|----------- + -----------|*\- e + e /*\e + e /
\ 2 2 /
---------------------------------------------------------------------------------------------
/ / 3*pi\ / 3*pi\\
| I*|-a + ----| I*|a - ----||
| \ 2 / \ 2 /|
2*\- e + e /
$$- \frac{\left(\frac{e^{i \left(- a + \pi\right)}}{2} + \frac{e^{i \left(a - \pi\right)}}{2}\right) \left(- e^{i \left(- a - \pi\right)} + e^{i \left(a + \pi\right)}\right) \left(e^{i \left(- a + \frac{3 \pi}{2}\right)} + e^{i \left(a - \frac{3 \pi}{2}\right)}\right)}{2 \left(- e^{i \left(- a + \frac{3 \pi}{2}\right)} + e^{i \left(a - \frac{3 \pi}{2}\right)}\right)}$$
-(exp(i*(pi - a))/2 + exp(i*(a - pi))/2)*(-exp(i*(-pi - a)) + exp(i*(pi + a)))*(exp(i*(a - 3*pi/2)) + exp(i*(-a + 3*pi/2)))/(2*(-exp(i*(-a + 3*pi/2)) + exp(i*(a - 3*pi/2))))
Тригонометрическая часть
[src]
2 sin (a)
$$\sin^{2}{\left(a \right)}$$
1 ------- 2 csc (a)
$$\frac{1}{\csc^{2}{\left(a \right)}}$$
2/ pi\
cos |a - --|
\ 2 /
$$\cos^{2}{\left(a - \frac{\pi}{2} \right)}$$
1 ------------ 2 csc (pi - a)
$$\frac{1}{\csc^{2}{\left(- a + \pi \right)}}$$
1 cos(2*a) - - -------- 2 2
$$- \frac{\cos{\left(2 a \right)}}{2} + \frac{1}{2}$$
1
------------
2/ pi\
sec |a - --|
\ 2 /
$$\frac{1}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}$$
1
------------
2/pi \
sec |-- - a|
\2 /
$$\frac{1}{\sec^{2}{\left(- a + \frac{\pi}{2} \right)}}$$
sin(2*a)*tan(a)
---------------
2
$$\frac{\sin{\left(2 a \right)} \tan{\left(a \right)}}{2}$$
2
tan (a)
-----------
2
1 + tan (a)
$$\frac{\tan^{2}{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1}$$
cos(a)*sin(a)
-------------
cot(a)
$$\frac{\sin{\left(a \right)} \cos{\left(a \right)}}{\cot{\left(a \right)}}$$
4/a\ 2/a\
- 4*cos |-| + 4*cos |-|
\2/ \2/
$$- 4 \cos^{4}{\left(\frac{a}{2} \right)} + 4 \cos^{2}{\left(\frac{a}{2} \right)}$$
3
2*sin (a)*cos(a)
----------------
sin(2*a)
$$\frac{2 \sin^{3}{\left(a \right)} \cos{\left(a \right)}}{\sin{\left(2 a \right)}}$$
2*csc(2*a) -------------- 3 csc (a)*sec(a)
$$\frac{2 \csc{\left(2 a \right)}}{\csc^{3}{\left(a \right)} \sec{\left(a \right)}}$$
/a\
(1 + cos(a))*sin(a)*tan|-|
\2/
$$\left(\cos{\left(a \right)} + 1\right) \sin{\left(a \right)} \tan{\left(\frac{a}{2} \right)}$$
/pi \
csc|-- - a|
\2 /
-----------------
2*csc(a)*csc(2*a)
$$\frac{\csc{\left(- a + \frac{\pi}{2} \right)}}{2 \csc{\left(a \right)} \csc{\left(2 a \right)}}$$
3 / pi\
2*sin (a)*sin|a + --|
\ 2 /
---------------------
sin(2*a)
$$\frac{2 \sin^{3}{\left(a \right)} \sin{\left(a + \frac{\pi}{2} \right)}}{\sin{\left(2 a \right)}}$$
2/a\
4*cot |-|
\2/
--------------
2
/ 2/a\\
|1 + cot |-||
\ \2//
$$\frac{4 \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}$$
2/a\
4*tan |-|
\2/
--------------
2
/ 2/a\\
|1 + tan |-||
\ \2//
$$\frac{4 \tan^{2}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}$$
/ 0 for a mod pi = 0 | < 2 |sin (a) otherwise \
$$\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin^{2}{\left(a \right)} & \text{otherwise} \end{cases}$$
sec(a)
---------------------------
/ pi\ / pi\
2*sec|a - --|*sec|2*a - --|
\ 2 / \ 2 /
$$\frac{\sec{\left(a \right)}}{2 \sec{\left(a - \frac{\pi}{2} \right)} \sec{\left(2 a - \frac{\pi}{2} \right)}}$$
/ pi\ / pi\
cos|a - --|*cos|2*a - --|
\ 2 / \ 2 /
-------------------------
2*cos(a)
$$\frac{\cos{\left(a - \frac{\pi}{2} \right)} \cos{\left(2 a - \frac{\pi}{2} \right)}}{2 \cos{\left(a \right)}}$$
3/ pi\
2*cos |a - --|*cos(a)
\ 2 /
---------------------
/ pi\
cos|2*a - --|
\ 2 /
$$\frac{2 \cos{\left(a \right)} \cos^{3}{\left(a - \frac{\pi}{2} \right)}}{\cos{\left(2 a - \frac{\pi}{2} \right)}}$$
/ pi\
2*sec|2*a - --|
\ 2 /
-------------------
3/ pi\
sec(a)*sec |a - --|
\ 2 /
$$\frac{2 \sec{\left(2 a - \frac{\pi}{2} \right)}}{\sec{\left(a \right)} \sec^{3}{\left(a - \frac{\pi}{2} \right)}}$$
2
/ 2/a pi\\ 2
|1 - cot |- + --|| *(1 + sin(a))
\ \2 4 //
---------------------------------
4
$$\frac{\left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\sin{\left(a \right)} + 1\right)^{2}}{4}$$
// 0 for 2*a mod pi = 0\
|< |*tan(a)
\\sin(2*a) otherwise /
--------------------------------------
2
$$\frac{\left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\sin{\left(2 a \right)} & \text{otherwise} \end{cases}\right) \tan{\left(a \right)}}{2}$$
2
/ 2/a pi\\
|-1 + tan |- + --||
\ \2 4 //
--------------------
2
/ 2/a pi\\
|1 + tan |- + --||
\ \2 4 //
$$\frac{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
2
/ 2/a pi\\
|1 - cot |- + --||
\ \2 4 //
-------------------
2
/ 2/a pi\\
|1 + cot |- + --||
\ \2 4 //
$$\frac{\left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
2/a\ 2/a\
-1 + 2*cos |-| - 2*cos |-|*cos(2*a) + cos(2*a)
\2/ \2/
----------------------------------------------
2*cos(a)
$$\frac{- 2 \cos^{2}{\left(\frac{a}{2} \right)} \cos{\left(2 a \right)} + 2 \cos^{2}{\left(\frac{a}{2} \right)} + \cos{\left(2 a \right)} - 1}{2 \cos{\left(a \right)}}$$
/ 0 for 2*a mod pi = 0
|
| 2*cot(a)
<----------- otherwise
| 2
|1 + cot (a)
\
--------------------------------
2*cot(a)
$$\frac{\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{2 \cot{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}}{2 \cot{\left(a \right)}}$$
/ 2/a\\ /a\
2*|1 - tan |-||*tan(a)*tan|-|
\ \2// \2/
-----------------------------
2
/ 2/a\\
|1 + tan |-||
\ \2//
$$\frac{2 \cdot \left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \tan{\left(\frac{a}{2} \right)} \tan{\left(a \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}$$
/ 0 for a mod pi = 0 | | 2/a\ | 4*cot |-| | \2/ <-------------- otherwise | 2 |/ 2/a\\ ||1 + cot |-|| |\ \2// \
$$\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}$$
/ / 3*pi\ | 1 for |a + ----| mod 2*pi = 0 | \ 2 / < | 4/a\ 2/a\ |- 4*cos |-| + 4*cos |-| otherwise \ \2/ \2/
$$\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\- 4 \cos^{4}{\left(\frac{a}{2} \right)} + 4 \cos^{2}{\left(\frac{a}{2} \right)} & \text{otherwise} \end{cases}$$
/ 1 \
2*|1 - -------|
| 2/a\|
| cot |-||
\ \2//
----------------------------
2
/ 1 \ /a\
|1 + -------| *cot(a)*cot|-|
| 2/a\| \2/
| cot |-||
\ \2//
$$\frac{2 \cdot \left(1 - \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right)}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right)^{2} \cot{\left(\frac{a}{2} \right)} \cot{\left(a \right)}}$$
// 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\ |< |*|< |*tan(a) \\sin(a) otherwise / \\cos(a) otherwise /
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right) \tan{\left(a \right)}$$
/ / 3*pi\ | 1 for |a + ----| mod 2*pi = 0 | \ 2 / | | 2 |/ 2/a pi\\ <|-1 + tan |- + --|| |\ \2 4 // |-------------------- otherwise | 2 |/ 2/a pi\\ ||1 + tan |- + --|| \\ \2 4 //
$$\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}$$
/ 2 \
2 2/a\ |sin (a) 4/a\|
8*sin (a)*sin |-|*|------- - 2*sin |-||
\2/ \ 2 \2//
---------------------------------------
2
/ 2 4/a\\
|sin (a) + 4*sin |-|| *cos(a)
\ \2//
$$\frac{8 \left(- 2 \sin^{4}{\left(\frac{a}{2} \right)} + \frac{\sin^{2}{\left(a \right)}}{2}\right) \sin^{2}{\left(\frac{a}{2} \right)} \sin^{2}{\left(a \right)}}{\left(4 \sin^{4}{\left(\frac{a}{2} \right)} + \sin^{2}{\left(a \right)}\right)^{2} \cos{\left(a \right)}}$$
/ 4/a\\
| 4*sin |-||
2/a\ | \2/|
8*sin |-|*|1 - ---------|*sin(a)
\2/ | 2 |
\ sin (a) /
--------------------------------
2
/ 4/a\\
| 4*sin |-||
| \2/|
|1 + ---------| *sin(2*a)
| 2 |
\ sin (a) /
$$\frac{8 \left(- \frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right) \sin^{2}{\left(\frac{a}{2} \right)} \sin{\left(a \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right)^{2} \sin{\left(2 a \right)}}$$
3/a\ / 2 \ /a pi\
16*tan |-|*\1 + tan (a)/*tan|- + --|
\2/ \2 4 /
----------------------------------------
3
/ 2/a\\ / 2/a pi\\
|1 + tan |-|| *|1 + tan |- + --||*tan(a)
\ \2// \ \2 4 //
$$\frac{16 \left(\tan^{2}{\left(a \right)} + 1\right) \tan^{3}{\left(\frac{a}{2} \right)} \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{3} \left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \tan{\left(a \right)}}$$
// 0 for a mod pi = 0\ || | ||1 - cos(a) | // 1 for a mod 2*pi = 0\ |<---------- otherwise |*|< |*tan(a) || /a\ | \\cos(a) otherwise / || tan|-| | \\ \2/ /
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{- \cos{\left(a \right)} + 1}{\tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right) \tan{\left(a \right)}$$
// 1 for a mod 2*pi = 0\
2 // 0 for a mod pi = 0\ || |
2*sin (a)*|< |*|< / pi\ |
\\sin(a) otherwise / ||sin|a + --| otherwise |
\\ \ 2 / /
------------------------------------------------------------------------
sin(2*a)
$$\frac{2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\sin{\left(a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \sin^{2}{\left(a \right)}}{\sin{\left(2 a \right)}}$$
// 0 for a mod pi = 0\
|| | // 1 for a mod 2*pi = 0\ / pi\
|< / pi\ |*|< |*cos|a - --|
||cos|a - --| otherwise | \\cos(a) otherwise / \ 2 /
\\ \ 2 / /
--------------------------------------------------------------------------
cos(a)
$$\frac{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right) \cos{\left(a - \frac{\pi}{2} \right)}}{\cos{\left(a \right)}}$$
// 0 for a mod pi = 0\
|| | // 1 for a mod 2*pi = 0\
|| 1 | || |
|<----------- otherwise |*|< 1 |*sec(a)
|| / pi\ | ||------ otherwise |
||sec|a - --| | \\sec(a) /
\\ \ 2 / /
---------------------------------------------------------------------
/ pi\
sec|a - --|
\ 2 /
$$\frac{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(a \right)}} & \text{otherwise} \end{cases}\right) \sec{\left(a \right)}}{\sec{\left(a - \frac{\pi}{2} \right)}}$$
// 1 for a mod 2*pi = 0\
// 0 for a mod pi = 0\ || |
|| | || 1 | /pi \
|< 1 |*|<----------- otherwise |*csc|-- - a|
||------ otherwise | || /pi \ | \2 /
\\csc(a) / ||csc|-- - a| |
\\ \2 / /
--------------------------------------------------------------------------
csc(a)
$$\frac{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\csc{\left(a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \csc{\left(- a + \frac{\pi}{2} \right)}}{\csc{\left(a \right)}}$$
/ 2/a\ \
| sec |-| |
| \2/ | /a\
2*|1 - ------------|*sec(a)*sec|-|
| 2/a pi\| \2/
| sec |- - --||
\ \2 2 //
-------------------------------------------
2
/ 2/a\ \
| sec |-| |
| \2/ | / pi\ /a pi\
|1 + ------------| *sec|a - --|*sec|- - --|
| 2/a pi\| \ 2 / \2 2 /
| sec |- - --||
\ \2 2 //
$$\frac{2 \left(- \frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{a}{2} \right)} \sec{\left(a \right)}}{\left(\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)} \sec{\left(a - \frac{\pi}{2} \right)}}$$
// 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\ || | || | || /a\ | || 2/a\ | || 2*tan|-| | ||1 - tan |-| | |< \2/ |*|< \2/ |*tan(a) ||----------- otherwise | ||----------- otherwise | || 2/a\ | || 2/a\ | ||1 + tan |-| | ||1 + tan |-| | \\ \2/ / \\ \2/ /
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{a}{2} \right)} + 1}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \tan{\left(a \right)}$$
/ 2/a pi\\
| cos |- - --||
| \2 2 /| / pi\ /a pi\
2*|1 - ------------|*cos|a - --|*cos|- - --|
| 2/a\ | \ 2 / \2 2 /
| cos |-| |
\ \2/ /
--------------------------------------------
2
/ 2/a pi\\
| cos |- - --||
| \2 2 /| /a\
|1 + ------------| *cos(a)*cos|-|
| 2/a\ | \2/
| cos |-| |
\ \2/ /
$$\frac{2 \cdot \left(1 - \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right) \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)} \cos{\left(a - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right)^{2} \cos{\left(\frac{a}{2} \right)} \cos{\left(a \right)}}$$
// 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/ /
--------------------------------------------------------------------
cot(a)
$$\frac{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)}{\cot{\left(a \right)}}$$
/ 2/pi a\\
| csc |-- - -||
| \2 2/| /pi \ /pi a\
2*|1 - ------------|*csc|-- - a|*csc|-- - -|
| 2/a\ | \2 / \2 2/
| csc |-| |
\ \2/ /
--------------------------------------------
2
/ 2/pi a\\
| csc |-- - -||
| \2 2/| /a\
|1 + ------------| *csc(a)*csc|-|
| 2/a\ | \2/
| csc |-| |
\ \2/ /
$$\frac{2 \cdot \left(1 - \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right) \csc{\left(- a + \frac{\pi}{2} \right)} \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right)^{2} \csc{\left(\frac{a}{2} \right)} \csc{\left(a \right)}}$$
// 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\
|| | || |
|
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \tan{\left(a \right)}$$
// 1 for a mod 2*pi = 0\
|| |
// 0 for a mod pi = 0\ || 1 |
|| | ||-1 + ------- |
|| 2 | || 2/a\ |
||-------------------- otherwise | || tan |-| |
|
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right) \tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right) \tan{\left(a \right)}$$
// / pi\ \
// 0 for a mod pi = 0\ || 0 for |a + --| mod pi = 0| // zoo for 2*a mod pi = 0\
|| | || \ 2 / | || |
2*|<-sin(3*a) + 3*sin(a) |*|< |*|< 1 |
||-------------------- otherwise | || /a pi\ | ||-------- otherwise |
\\ 4 / ||(1 + sin(a))*cot|- + --| otherwise | \\sin(2*a) /
\\ \2 4 / /
$$2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{3 \sin{\left(a \right)} - \sin{\left(3 a \right)}}{4} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(a \right)} + 1\right) \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: 2 a \bmod \pi = 0 \\\frac{1}{\sin{\left(2 a \right)}} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\ || | || | || 2*(-sin(2*a) + 2*sin(a)) | || -2 - 2*cos(2*a) + 4*cos(a) | |<------------------------------ otherwise |*|<------------------------------ otherwise |*tan(a) || 2 | || 2 | ||1 - cos(2*a) + 2*(1 - cos(a)) | ||1 - cos(2*a) + 2*(1 - cos(a)) | \\ / \\ /
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cdot \left(2 \sin{\left(a \right)} - \sin{\left(2 a \right)}\right)}{2 \left(- \cos{\left(a \right)} + 1\right)^{2} - \cos{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{4 \cos{\left(a \right)} - 2 \cos{\left(2 a \right)} - 2}{2 \left(- \cos{\left(a \right)} + 1\right)^{2} - \cos{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}\right) \tan{\left(a \right)}$$
// 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/ /
------------------------------------------------------------------------------------------------------------
cot(a)
$$\frac{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)}{\cot{\left(a \right)}}$$
// 1 for a mod 2*pi = 0\
|| |
// 0 for a mod pi = 0\ || 2 |
|| | || sin (a) |
|| sin(a) | ||-1 + --------- |
||----------------------- otherwise | || 4/a\ |
2 ||/ 2 \ | || 4*sin |-| |
2*sin (a)*|<| sin (a) | 2/a\ |*|< \2/ |
|||1 + ---------|*sin |-| | ||-------------- otherwise |
||| 4/a\| \2/ | || 2 |
||| 4*sin |-|| | || sin (a) |
||\ \2// | ||1 + --------- |
\\ / || 4/a\ |
|| 4*sin |-| |
\\ \2/ /
--------------------------------------------------------------------------------------------
sin(2*a)
$$\frac{2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{\sin{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \sin^{2}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}}{1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right) \sin^{2}{\left(a \right)}}{\sin{\left(2 a \right)}}$$
// 0 for a mod pi = 0\ // / pi\ \ || | || 0 for |a + --| mod pi = 0| || 3/a\ | || \ 2 / | // zoo for 2*a mod pi = 0\ || 8*cot |-| | || | || | || \2/ | || /a pi\ | || 2 | 2*|<-------------- otherwise |*|< 2*cot|- + --| |*|<1 + cot (a) | || 3 | || \2 4 / | ||----------- otherwise | ||/ 2/a\\ | ||---------------- otherwise | || 2*cot(a) | |||1 + cot |-|| | || 2/a pi\ | \\ / ||\ \2// | ||1 + cot |- + --| | \\ / \\ \2 4 / /
$$2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{8 \cot^{3}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{3}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: 2 a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} + 1}{2 \cot{\left(a \right)}} & \text{otherwise} \end{cases}\right)$$
// 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 |- - --| | / pi\
|
$$\frac{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{a}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) \cos{\left(a - \frac{\pi}{2} \right)}}{\cos{\left(a \right)}}$$
// 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 |-| |
|
$$\frac{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right) \sec{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right) \sec{\left(a \right)}}{\sec{\left(a - \frac{\pi}{2} \right)}}$$
// 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 |-- - -| | /pi \
|
$$\frac{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{a}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) \csc{\left(- a + \frac{\pi}{2} \right)}}{\csc{\left(a \right)}}$$
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))*csc(pi/2 - a)/csc(a)