Господин Экзамен
Общий знаменатель (sin((pi/2)-a)*tan(pi+2))/(cos((3*pi/2)+a))
Решение
Вы ввели
[src]
/pi \
sin|-- - a|*tan(pi + 2)
\2 /
-----------------------
/3*pi \
cos|---- + a|
\ 2 /
$$\frac{\sin{\left(- a + \frac{\pi}{2} \right)} \tan{\left(2 + \pi \right)}}{\cos{\left(a + \frac{3 \pi}{2} \right)}}$$
sin(pi/2 - a)*tan(pi + 2)/cos(3*pi/2 + a)
Общее упрощение
[src]
tan(2) ------ tan(a)
$$\frac{\tan{\left(2 \right)}}{\tan{\left(a \right)}}$$
tan(2)/tan(a)
Численный ответ
[src]
-2.18503986326152*sin(pi/2 - a)/cos(3*pi/2 + a)
-2.18503986326152*sin(pi/2 - a)/cos(3*pi/2 + a)
Степени
[src]
cos(a)*tan(2)
-------------
sin(a)
$$\frac{\cos{\left(a \right)} \tan{\left(2 \right)}}{\sin{\left(a \right)}}$$
cos(a)*tan(pi + 2)
------------------
sin(a)
$$\frac{\cos{\left(a \right)} \tan{\left(2 + \pi \right)}}{\sin{\left(a \right)}}$$
cos(a)*tan(pi + 2)
------------------
/3*pi \
cos|---- + a|
\ 2 /
$$\frac{\cos{\left(a \right)} \tan{\left(2 + \pi \right)}}{\cos{\left(a + \frac{3 \pi}{2} \right)}}$$
/ / pi\ /pi \\
| I*|a - --| I*|-- - a||
/ I*(2 + pi) I*(-2 - pi)\ | \ 2 / \2 /|
\- e + e /*\- e + e /
---------------------------------------------------------------
/ / 3*pi\ / 3*pi\\
| I*|a + ----| I*|-a - ----||
| \ 2 / \ 2 /|
|e e | / I*(-2 - pi) I*(2 + pi)\
2*|------------- + --------------|*\e + e /
\ 2 2 /
$$\frac{\left(- e^{i \left(2 + \pi\right)} + e^{i \left(- \pi - 2\right)}\right) \left(e^{i \left(- a + \frac{\pi}{2}\right)} - e^{i \left(a - \frac{\pi}{2}\right)}\right)}{2 \left(e^{i \left(2 + \pi\right)} + e^{i \left(- \pi - 2\right)}\right) \left(\frac{e^{i \left(- a - \frac{3 \pi}{2}\right)}}{2} + \frac{e^{i \left(a + \frac{3 \pi}{2}\right)}}{2}\right)}$$
(-exp(i*(2 + pi)) + exp(i*(-2 - pi)))*(-exp(i*(a - pi/2)) + exp(i*(pi/2 - a)))/(2*(exp(i*(a + 3*pi/2))/2 + exp(i*(-a - 3*pi/2))/2)*(exp(i*(-2 - pi)) + exp(i*(2 + pi))))
Рациональный знаменатель
[src]
cos(a)*tan(2)
-------------
sin(a)
$$\frac{\cos{\left(a \right)} \tan{\left(2 \right)}}{\sin{\left(a \right)}}$$
cos(a)*tan(2)/sin(a)
Общий знаменатель
[src]
cos(a)*tan(2)
-------------
sin(a)
$$\frac{\cos{\left(a \right)} \tan{\left(2 \right)}}{\sin{\left(a \right)}}$$
cos(a)*tan(2)/sin(a)
Комбинаторика
[src]
cos(a)*tan(2)
-------------
sin(a)
$$\frac{\cos{\left(a \right)} \tan{\left(2 \right)}}{\sin{\left(a \right)}}$$
cos(a)*tan(2)/sin(a)
Объединение рациональных выражений
[src]
/pi - 2*a\
sin|--------|*tan(2)
\ 2 /
--------------------
/2*a + 3*pi\
cos|----------|
\ 2 /
$$\frac{\sin{\left(\frac{- 2 a + \pi}{2} \right)} \tan{\left(2 \right)}}{\cos{\left(\frac{2 a + 3 \pi}{2} \right)}}$$
sin((pi - 2*a)/2)*tan(2)/cos((2*a + 3*pi)/2)
Собрать выражение
[src]
cos(a)*csc(a)*tan(2)
$$\cos{\left(a \right)} \tan{\left(2 \right)} \csc{\left(a \right)}$$
cos(a)*csc(a)*tan(2)
Тригонометрическая часть
[src]
cot(a)*tan(2)
$$\tan{\left(2 \right)} \cot{\left(a \right)}$$
cot(a) ------ cot(2)
$$\frac{\cot{\left(a \right)}}{\cot{\left(2 \right)}}$$
tan(2) ------ tan(a)
$$\frac{\tan{\left(2 \right)}}{\tan{\left(a \right)}}$$
2 2*sin (2) ------------- sin(4)*tan(a)
$$\frac{2 \sin^{2}{\left(2 \right)}}{\sin{\left(4 \right)} \tan{\left(a \right)}}$$
cos(a)*tan(pi + 2)
------------------
sin(a)
$$\frac{\cos{\left(a \right)} \tan{\left(2 + \pi \right)}}{\sin{\left(a \right)}}$$
csc(a)*sec(2) ------------- csc(2)*sec(a)
$$\frac{\csc{\left(a \right)} \sec{\left(2 \right)}}{\csc{\left(2 \right)} \sec{\left(a \right)}}$$
cos(a)*sin(2) ------------- cos(2)*sin(a)
$$\frac{\sin{\left(2 \right)} \cos{\left(a \right)}}{\sin{\left(a \right)} \cos{\left(2 \right)}}$$
2 2*sin (2)*cos(a) ---------------- sin(4)*sin(a)
$$\frac{2 \sin^{2}{\left(2 \right)} \cos{\left(a \right)}}{\sin{\left(4 \right)} \sin{\left(a \right)}}$$
2*csc(4)*csc(a)
---------------
2
csc (2)*sec(a)
$$\frac{2 \csc{\left(4 \right)} \csc{\left(a \right)}}{\csc^{2}{\left(2 \right)} \sec{\left(a \right)}}$$
2
sin (2)*sin(2*a)
----------------
2
sin(4)*sin (a)
$$\frac{\sin^{2}{\left(2 \right)} \sin{\left(2 a \right)}}{\sin{\left(4 \right)} \sin^{2}{\left(a \right)}}$$
1 /a\
- ------ + cot|-|
/a\ \2/
cot|-|
\2/
-----------------
2*cot(2)
$$\frac{\cot{\left(\frac{a}{2} \right)} - \frac{1}{\cot{\left(\frac{a}{2} \right)}}}{2 \cot{\left(2 \right)}}$$
/ /a\ /a\\
|- tan|-| + cot|-||*tan(2)
\ \2/ \2//
--------------------------
2
$$\frac{\left(- \tan{\left(\frac{a}{2} \right)} + \cot{\left(\frac{a}{2} \right)}\right) \tan{\left(2 \right)}}{2}$$
2 / pi\
2*sin (2)*sin|a + --|
\ 2 /
---------------------
sin(4)*sin(a)
$$\frac{2 \sin^{2}{\left(2 \right)} \sin{\left(a + \frac{\pi}{2} \right)}}{\sin{\left(4 \right)} \sin{\left(a \right)}}$$
-sin(-2 + a) + sin(2 + a) ------------------------- sin(-2 + a) + sin(2 + a)
$$\frac{- \sin{\left(a - 2 \right)} + \sin{\left(a + 2 \right)}}{\sin{\left(a - 2 \right)} + \sin{\left(a + 2 \right)}}$$
/ 2/a\\
|1 - tan |-||*tan(2)
\ \2//
--------------------
/a\
2*tan|-|
\2/
$$\frac{\left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \tan{\left(2 \right)}}{2 \tan{\left(\frac{a}{2} \right)}}$$
2*csc(4)*csc(a)
-------------------
2 /pi \
csc (2)*csc|-- - a|
\2 /
$$\frac{2 \csc{\left(4 \right)} \csc{\left(a \right)}}{\csc^{2}{\left(2 \right)} \csc{\left(- a + \frac{\pi}{2} \right)}}$$
/ 1 /a\\
|------ - tan|-||*tan(2)
| /a\ \2/|
|tan|-| |
\ \2/ /
------------------------
2
$$\frac{\left(- \tan{\left(\frac{a}{2} \right)} + \frac{1}{\tan{\left(\frac{a}{2} \right)}}\right) \tan{\left(2 \right)}}{2}$$
cos(a)*tan(2)
-------------------
/a\
(1 + cos(a))*tan|-|
\2/
$$\frac{\cos{\left(a \right)} \tan{\left(2 \right)}}{\left(\cos{\left(a \right)} + 1\right) \tan{\left(\frac{a}{2} \right)}}$$
2
8*tan (1)
----------------------------
2
/ 2 \
\1 + tan (1)/ *sin(4)*tan(a)
$$\frac{8 \tan^{2}{\left(1 \right)}}{\left(1 + \tan^{2}{\left(1 \right)}\right)^{2} \sin{\left(4 \right)} \tan{\left(a \right)}}$$
/ 1 \ /a\
|1 - -------|*cot|-|
| 2/a\| \2/
| cot |-||
\ \2//
--------------------
2*cot(2)
$$\frac{\left(1 - \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \cot{\left(\frac{a}{2} \right)}}{2 \cot{\left(2 \right)}}$$
/ pi\
sin(2)*sin|a + --|
\ 2 /
------------------
/ pi\
sin(a)*sin|2 + --|
\ 2 /
$$\frac{\sin{\left(2 \right)} \sin{\left(a + \frac{\pi}{2} \right)}}{\sin{\left(a \right)} \sin{\left(\frac{\pi}{2} + 2 \right)}}$$
/ pi\
cos(a)*cos|2 - --|
\ 2 /
------------------
/ pi\
cos(2)*cos|a - --|
\ 2 /
$$\frac{\cos{\left(a \right)} \cos{\left(- \frac{\pi}{2} + 2 \right)}}{\cos{\left(2 \right)} \cos{\left(a - \frac{\pi}{2} \right)}}$$
/ pi\
sec(2)*sec|a - --|
\ 2 /
------------------
/ pi\
sec(a)*sec|2 - --|
\ 2 /
$$\frac{\sec{\left(2 \right)} \sec{\left(a - \frac{\pi}{2} \right)}}{\sec{\left(a \right)} \sec{\left(- \frac{\pi}{2} + 2 \right)}}$$
/ pi\
-cos(a)*cos|2 + --|
\ 2 /
--------------------
/ pi\
cos(2)*cos|a - --|
\ 2 /
$$- \frac{\cos{\left(a \right)} \cos{\left(\frac{\pi}{2} + 2 \right)}}{\cos{\left(2 \right)} \cos{\left(a - \frac{\pi}{2} \right)}}$$
/pi \
sec(2)*sec|-- - a|
\2 /
-------------------
/ pi\
sec(a)*sec|-2 + --|
\ 2 /
$$\frac{\sec{\left(2 \right)} \sec{\left(- a + \frac{\pi}{2} \right)}}{\sec{\left(a \right)} \sec{\left(-2 + \frac{\pi}{2} \right)}}$$
/ pi\
csc(a)*csc|-2 + --|
\ 2 /
-------------------
/pi \
csc(2)*csc|-- - a|
\2 /
$$\frac{\csc{\left(a \right)} \csc{\left(-2 + \frac{\pi}{2} \right)}}{\csc{\left(2 \right)} \csc{\left(- a + \frac{\pi}{2} \right)}}$$
/ pi\
-sin(2 + pi)*sin|a + --|
\ 2 /
-------------------------
/ pi\
sin(a)*sin|2 + --|
\ 2 /
$$- \frac{\sin{\left(2 + \pi \right)} \sin{\left(a + \frac{\pi}{2} \right)}}{\sin{\left(a \right)} \sin{\left(\frac{\pi}{2} + 2 \right)}}$$
/ pi\
csc|-2 + --|*csc(pi - a)
\ 2 /
------------------------
/pi \
csc(-2 + pi)*csc|-- - a|
\2 /
$$\frac{\csc{\left(-2 + \frac{\pi}{2} \right)} \csc{\left(- a + \pi \right)}}{\csc{\left(-2 + \pi \right)} \csc{\left(- a + \frac{\pi}{2} \right)}}$$
/ 1 /a\\
|- ------ + tan|-||*tan(1)
| /a\ \2/|
| tan|-| |
\ \2/ /
--------------------------
2
-1 + tan (1)
$$\frac{\left(\tan{\left(\frac{a}{2} \right)} - \frac{1}{\tan{\left(\frac{a}{2} \right)}}\right) \tan{\left(1 \right)}}{-1 + \tan^{2}{\left(1 \right)}}$$
2/ pi\
2*cos |2 - --|*cos(a)
\ 2 /
-----------------------
/ pi\ / pi\
cos|4 - --|*cos|a - --|
\ 2 / \ 2 /
$$\frac{2 \cos{\left(a \right)} \cos^{2}{\left(- \frac{\pi}{2} + 2 \right)}}{\cos{\left(- \frac{\pi}{2} + 4 \right)} \cos{\left(a - \frac{\pi}{2} \right)}}$$
/ pi\ / pi\
2*sec|4 - --|*sec|a - --|
\ 2 / \ 2 /
-------------------------
2/ pi\
sec(a)*sec |2 - --|
\ 2 /
$$\frac{2 \sec{\left(- \frac{\pi}{2} + 4 \right)} \sec{\left(a - \frac{\pi}{2} \right)}}{\sec{\left(a \right)} \sec^{2}{\left(- \frac{\pi}{2} + 2 \right)}}$$
/ 2/a\\
|1 - tan |-||*tan(1)
\ \2//
--------------------
/ 2 \ /a\
\1 - tan (1)/*tan|-|
\2/
$$\frac{\left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \tan{\left(1 \right)}}{\left(- \tan^{2}{\left(1 \right)} + 1\right) \tan{\left(\frac{a}{2} \right)}}$$
2 / 2 \
4*cot (1)*\1 + cot (2)/*cot(a)
------------------------------
2
/ 2 \
\1 + cot (1)/ *cot(2)
$$\frac{4 \left(\cot^{2}{\left(2 \right)} + 1\right) \cot^{2}{\left(1 \right)} \cot{\left(a \right)}}{\left(\cot^{2}{\left(1 \right)} + 1\right)^{2} \cot{\left(2 \right)}}$$
2 / 2 \
4*tan (1)*\1 + tan (2)/
----------------------------
2
/ 2 \
\1 + tan (1)/ *tan(2)*tan(a)
$$\frac{4 \cdot \left(1 + \tan^{2}{\left(2 \right)}\right) \tan^{2}{\left(1 \right)}}{\left(1 + \tan^{2}{\left(1 \right)}\right)^{2} \tan{\left(2 \right)} \tan{\left(a \right)}}$$
/ 2/a\\
| 2*sin |-||
2 | sin(a) \2/|
sin (2)*|---------- - ---------|
\1 - cos(a) sin(a) /
--------------------------------
sin(4)
$$\frac{\left(- \frac{2 \sin^{2}{\left(\frac{a}{2} \right)}}{\sin{\left(a \right)}} + \frac{\sin{\left(a \right)}}{- \cos{\left(a \right)} + 1}\right) \sin^{2}{\left(2 \right)}}{\sin{\left(4 \right)}}$$
/ 4/a\\
| 4*sin |-||
2 | \2/|
sin (2)*|-sin(a) + ---------|
\ sin(a) /
-----------------------------
(-1 + cos(a))*sin(4)
$$\frac{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin{\left(a \right)}} - \sin{\left(a \right)}\right) \sin^{2}{\left(2 \right)}}{\left(\cos{\left(a \right)} - 1\right) \sin{\left(4 \right)}}$$
/ 4/a\\
| 4*sin |-||
2 | \2/|
sin (2)*|1 - ---------|*sin(a)
| 2 |
\ sin (a) /
------------------------------
2/a\
2*sin(4)*sin |-|
\2/
$$\frac{\left(- \frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right) \sin^{2}{\left(2 \right)} \sin{\left(a \right)}}{2 \sin{\left(4 \right)} \sin^{2}{\left(\frac{a}{2} \right)}}$$
/ 2/a\\
| 2*sin |-||
2 | sin(a) \2/|
sin (2)*|--------- - ---------|
| 2/a\ sin(a) |
|2*sin |-| |
\ \2/ /
-------------------------------
sin(4)
$$\frac{\left(- \frac{2 \sin^{2}{\left(\frac{a}{2} \right)}}{\sin{\left(a \right)}} + \frac{\sin{\left(a \right)}}{2 \sin^{2}{\left(\frac{a}{2} \right)}}\right) \sin^{2}{\left(2 \right)}}{\sin{\left(4 \right)}}$$
2 / 2/a\\
16*tan (1)*|-1 + 2*cos |-||
\ \2//
------------------------------------------
2
/ 2 \
\1 + tan (1)/ *(-cos(4 + a) + cos(-4 + a))
$$\frac{16 \cdot \left(2 \cos^{2}{\left(\frac{a}{2} \right)} - 1\right) \tan^{2}{\left(1 \right)}}{\left(1 + \tan^{2}{\left(1 \right)}\right)^{2} \left(\cos{\left(a - 4 \right)} - \cos{\left(a + 4 \right)}\right)}$$
/ /a\ /a pi\\
| cos|-| cos|- - --||
| \2/ \2 2 /| / pi\
|----------- - -----------|*cos|2 - --|
| /a pi\ /a\ | \ 2 /
|cos|- - --| cos|-| |
\ \2 2 / \2/ /
---------------------------------------
2*cos(2)
$$\frac{\left(\frac{\cos{\left(\frac{a}{2} \right)}}{\cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - \frac{\cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos{\left(\frac{a}{2} \right)}}\right) \cos{\left(- \frac{\pi}{2} + 2 \right)}}{2 \cos{\left(2 \right)}}$$
/ 2/a pi\\
| cos |- - --||
| \2 2 /| /a\ / pi\
|1 - ------------|*cos|-|*cos|2 - --|
| 2/a\ | \2/ \ 2 /
| cos |-| |
\ \2/ /
-------------------------------------
/a pi\
2*cos(2)*cos|- - --|
\2 2 /
$$\frac{\left(1 - \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right) \cos{\left(\frac{a}{2} \right)} \cos{\left(- \frac{\pi}{2} + 2 \right)}}{2 \cos{\left(2 \right)} \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}$$
/ /a pi\ /a\ \
|sec|- - --| sec|-| |
| \2 2 / \2/ |
|----------- - -----------|*sec(2)
| /a\ /a pi\|
| sec|-| sec|- - --||
\ \2/ \2 2 //
----------------------------------
/ pi\
2*sec|2 - --|
\ 2 /
$$\frac{\left(- \frac{\sec{\left(\frac{a}{2} \right)}}{\sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + \frac{\sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec{\left(\frac{a}{2} \right)}}\right) \sec{\left(2 \right)}}{2 \sec{\left(- \frac{\pi}{2} + 2 \right)}}$$
/ /a\ /pi a\\
| csc|-| csc|-- - -||
| \2/ \2 2/| / pi\
|----------- - -----------|*csc|-2 + --|
| /pi a\ /a\ | \ 2 /
|csc|-- - -| csc|-| |
\ \2 2/ \2/ /
----------------------------------------
2*csc(2)
$$\frac{\left(\frac{\csc{\left(\frac{a}{2} \right)}}{\csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} - \frac{\csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc{\left(\frac{a}{2} \right)}}\right) \csc{\left(-2 + \frac{\pi}{2} \right)}}{2 \csc{\left(2 \right)}}$$
/ 2/a\ \
| sec |-| |
| \2/ | /a pi\
|1 - ------------|*sec(2)*sec|- - --|
| 2/a pi\| \2 2 /
| sec |- - --||
\ \2 2 //
-------------------------------------
/a\ / pi\
2*sec|-|*sec|2 - --|
\2/ \ 2 /
$$\frac{\left(- \frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(2 \right)} \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{2 \sec{\left(\frac{a}{2} \right)} \sec{\left(- \frac{\pi}{2} + 2 \right)}}$$
/ 2/pi a\\
| csc |-- - -||
| \2 2/| /a\ / pi\
|1 - ------------|*csc|-|*csc|-2 + --|
| 2/a\ | \2/ \ 2 /
| csc |-| |
\ \2/ /
--------------------------------------
/pi a\
2*csc(2)*csc|-- - -|
\2 2/
$$\frac{\left(1 - \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right) \csc{\left(\frac{a}{2} \right)} \csc{\left(-2 + \frac{\pi}{2} \right)}}{2 \csc{\left(2 \right)} \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}$$
// zoo for a mod pi = 0\
// 1 for a mod 2*pi = 0\ || |
|< |*|< 1 |*tan(2)
\\cos(a) otherwise / ||------ otherwise |
\\sin(a) /
$$\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sin{\left(a \right)}} & \text{otherwise} \end{cases}\right) \tan{\left(2 \right)}$$
// zoo for a mod pi = 0\
|| |
// 1 for a mod 2*pi = 0\ || /a\ |
|< |*|< -tan|-| |*tan(2)
\\cos(a) otherwise / || \2/ |
||----------- otherwise |
\\-1 + cos(a) /
$$\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\- \frac{\tan{\left(\frac{a}{2} \right)}}{\cos{\left(a \right)} - 1} & \text{otherwise} \end{cases}\right) \tan{\left(2 \right)}$$
// zoo for a mod pi = 0\
2 // 1 for a mod 2*pi = 0\ || |
2*sin (2)*|< |*|< 1 |
\\cos(a) otherwise / ||------ otherwise |
\\sin(a) /
-------------------------------------------------------------------
sin(4)
$$\frac{2 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sin{\left(a \right)}} & \text{otherwise} \end{cases}\right) \sin^{2}{\left(2 \right)}}{\sin{\left(4 \right)}}$$
// 1 for a mod 2*pi = 0\ // zoo for a mod pi = 0\
2 || | || |
2*sin (2)*|< / pi\ |*|< 1 |
||sin|a + --| otherwise | ||------ otherwise |
\\ \ 2 / / \\sin(a) /
------------------------------------------------------------------------
sin(4)
$$\frac{2 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\sin{\left(a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sin{\left(a \right)}} & \text{otherwise} \end{cases}\right) \sin^{2}{\left(2 \right)}}{\sin{\left(4 \right)}}$$
2 / 2/ pi\\ /a\ /a pi\
sin (1)*|1 + tan |1 + --||*cot(1)*tan|-|*tan|- + --|
\ \ 4 // \2/ \2 4 /
----------------------------------------------------
/ 2/a pi\\ 2/a\ / pi\
|1 + tan |- + --||*sin |-|*tan|1 + --|
\ \2 4 // \2/ \ 4 /
$$\frac{\left(1 + \tan^{2}{\left(\frac{\pi}{4} + 1 \right)}\right) \sin^{2}{\left(1 \right)} \tan{\left(\frac{a}{2} \right)} \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)} \cot{\left(1 \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \sin^{2}{\left(\frac{a}{2} \right)} \tan{\left(\frac{\pi}{4} + 1 \right)}}$$
// zoo for a mod pi = 0\
|| |
// 1 for a mod 2*pi = 0\ || 1 | / pi\
|< |*|<----------- otherwise |*cos|2 - --|
\\cos(a) otherwise / || / pi\ | \ 2 /
||cos|a - --| |
\\ \ 2 / /
--------------------------------------------------------------------------
cos(2)
$$\frac{\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\cos{\left(a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \cos{\left(- \frac{\pi}{2} + 2 \right)}}{\cos{\left(2 \right)}}$$
// 1 for a mod 2*pi = 0\ // zoo for a mod pi = 0\
|| | || |
|< 1 |*|< / pi\ |*sec(2)
||------ otherwise | ||sec|a - --| otherwise |
\\sec(a) / \\ \ 2 / /
---------------------------------------------------------------------
/ pi\
sec|2 - --|
\ 2 /
$$\frac{\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\sec{\left(a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \sec{\left(2 \right)}}{\sec{\left(- \frac{\pi}{2} + 2 \right)}}$$
// 1 for a mod 2*pi = 0\
|| |
|| 1 | // zoo for a mod pi = 0\ / pi\
|<----------- otherwise |*|< |*csc|-2 + --|
|| /pi \ | \\csc(a) otherwise / \ 2 /
||csc|-- - a| |
\\ \2 / /
---------------------------------------------------------------------------
csc(2)
$$\frac{\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\csc{\left(a \right)} & \text{otherwise} \end{cases}\right) \csc{\left(-2 + \frac{\pi}{2} \right)}}{\csc{\left(2 \right)}}$$
2 / 2 \ / 2/a\\ /a pi\
4*tan (1)*\1 + tan (2)/*|1 + tan |-||*tan|- + --|
\ \2// \2 4 /
-------------------------------------------------
2
/ 2 \ / 2/a pi\\ /a\
\1 + tan (1)/ *|1 + tan |- + --||*tan(2)*tan|-|
\ \2 4 // \2/
$$\frac{4 \cdot \left(1 + \tan^{2}{\left(2 \right)}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \tan^{2}{\left(1 \right)} \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(1 + \tan^{2}{\left(1 \right)}\right)^{2} \left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \tan{\left(2 \right)} \tan{\left(\frac{a}{2} \right)}}$$
// / pi\ \ || 0 for |a + --| mod pi = 0| || \ 2 / | // zoo for a mod pi = 0\ || | || | |< 1 + sin(a) |*|< 1 |*tan(2) ||----------- otherwise | ||------ otherwise | || /a pi\ | \\sin(a) / ||tan|- + --| | \\ \2 4 / /
$$\left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{\sin{\left(a \right)} + 1}{\tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sin{\left(a \right)}} & \text{otherwise} \end{cases}\right) \tan{\left(2 \right)}$$
/ 2/a\\ / 2/ pi\\ /a pi\
|1 + cot |-||*|1 + tan |1 + --||*cot(1)*tan|- + --|
\ \2// \ \ 4 // \2 4 /
---------------------------------------------------
/ 2 \ / 2/a pi\\ /a\ / pi\
\1 + cot (1)/*|1 + tan |- + --||*cot|-|*tan|1 + --|
\ \2 4 // \2/ \ 4 /
$$\frac{\left(1 + \tan^{2}{\left(\frac{\pi}{4} + 1 \right)}\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right) \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)} \cot{\left(1 \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(1 \right)} + 1\right) \tan{\left(\frac{\pi}{4} + 1 \right)} \cot{\left(\frac{a}{2} \right)}}$$
// 1 for a mod 2*pi = 0\ // zoo for a mod pi = 0\ || | || | || 2/a\ | || 2/a\ | ||1 - tan |-| | ||1 + tan |-| | |< \2/ |*|< \2/ |*tan(2) ||----------- otherwise | ||----------- otherwise | || 2/a\ | || /a\ | ||1 + tan |-| | || 2*tan|-| | \\ \2/ / \\ \2/ /
$$\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{a}{2} \right)} + 1}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{\tan^{2}{\left(\frac{a}{2} \right)} + 1}{2 \tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) \tan{\left(2 \right)}$$
// 1 for a mod 2*pi = 0\ // zoo for a mod pi = 0\
|| | || |
|| 2/a\ | || 2/a\ |
||-1 + cot |-| | ||1 + cot |-| |
|< \2/ |*|< \2/ |
||------------ otherwise | ||----------- otherwise |
|| 2/a\ | || /a\ |
||1 + cot |-| | || 2*cot|-| |
\\ \2/ / \\ \2/ /
--------------------------------------------------------------------
cot(2)
$$\frac{\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} + 1}{2 \cot{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)}{\cot{\left(2 \right)}}$$
2/a\ / 2 \ / 2/a\\ / 2/ pi\\
-cos |-|*\1 + tan (1)/*|1 - tan |-||*|1 - tan |1 + --||
\2/ \ \2// \ \ 4 //
----------------------------------------------------------------
/ 2/ pi\\ / 2/a pi\\ / 2 \ 2/a pi\
|1 + tan |1 + --||*|1 - cot |- + --||*\1 - tan (1)/*sin |- + --|
\ \ 4 // \ \2 4 // \2 4 /
$$- \frac{\left(1 + \tan^{2}{\left(1 \right)}\right) \left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(- \tan^{2}{\left(\frac{\pi}{4} + 1 \right)} + 1\right) \cos^{2}{\left(\frac{a}{2} \right)}}{\left(- \tan^{2}{\left(1 \right)} + 1\right) \left(1 + \tan^{2}{\left(\frac{\pi}{4} + 1 \right)}\right) \left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \sin^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
// 1 for a mod 2*pi = 0\ || | || 1 | // zoo for a mod pi = 0\ ||-1 + ------- | || | || 2/a\ | ||/ 1 \ /a\ | || tan |-| | |||1 + -------|*tan|-| | |< \2/ |*|<| 2/a\| \2/ |*tan(2) ||------------ otherwise | ||| tan |-|| | || 1 | ||\ \2// | ||1 + ------- | ||-------------------- otherwise | || 2/a\ | \\ 2 / || tan |-| | \\ \2/ /
$$\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right) \tan{\left(\frac{a}{2} \right)}}{2} & \text{otherwise} \end{cases}\right) \tan{\left(2 \right)}$$
// zoo for a mod pi = 0\
// 1 for a mod 2*pi = 0\ || |
|| | ||/ zoo for a mod pi = 0 |
|
$$\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sin{\left(a \right)}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \tan{\left(2 \right)}$$
/ 2 \ / 2/a pi\\ / 2/a\\ / 2/ pi\\
\1 + cot (1)/*|1 + tan |- + --||*|-1 + cot |-||*|-1 + tan |1 + --||
\ \2 4 // \ \2// \ \ 4 //
-------------------------------------------------------------------
/ 2/a\\ / 2/ pi\\ / 2 \ / 2/a pi\\
|1 + cot |-||*|1 + tan |1 + --||*\-1 + cot (1)/*|-1 + tan |- + --||
\ \2// \ \ 4 // \ \2 4 //
$$\frac{\left(-1 + \tan^{2}{\left(\frac{\pi}{4} + 1 \right)}\right) \left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(1 \right)} + 1\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)}{\left(-1 + \cot^{2}{\left(1 \right)}\right) \left(1 + \tan^{2}{\left(\frac{\pi}{4} + 1 \right)}\right) \left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)}$$
/ 2/a pi\\ / 2 \ / 2/a\\ / 2/ pi\\ -|1 + cot |- + --||*\1 + tan (1)/*|1 - tan |-||*|1 - tan |1 + --|| \ \2 4 // \ \2// \ \ 4 // ------------------------------------------------------------------- / 2/a\\ / 2/ pi\\ / 2/a pi\\ / 2 \ |1 + tan |-||*|1 + tan |1 + --||*|1 - cot |- + --||*\1 - tan (1)/ \ \2// \ \ 4 // \ \2 4 //
$$- \frac{\left(1 + \tan^{2}{\left(1 \right)}\right) \left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(- \tan^{2}{\left(\frac{\pi}{4} + 1 \right)} + 1\right) \left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)}{\left(- \tan^{2}{\left(1 \right)} + 1\right) \left(1 + \tan^{2}{\left(\frac{\pi}{4} + 1 \right)}\right) \left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}$$
// / 3*pi\ \
|| 1 for |a + ----| mod 2*pi = 0|
|| \ 2 / |
|| |
/ 2/ pi\\ // 1 for a mod 2*pi = 0\ || 1 /a\ |
-(1 + sin(2))*|-1 + cot |1 + --||*|< |*|<------ + tan|-| |
\ \ 4 // \\cos(a) otherwise / || /a\ \2/ |
||tan|-| |
|| \2/ |
||--------------- otherwise |
\\ 2 /
----------------------------------------------------------------------------------------------------------------
/ 2 \ 2
2*\-1 + cot (1)/*sin (1)
$$- \frac{\left(-1 + \cot^{2}{\left(\frac{\pi}{4} + 1 \right)}\right) \left(\sin{\left(2 \right)} + 1\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan{\left(\frac{a}{2} \right)} + \frac{1}{\tan{\left(\frac{a}{2} \right)}}}{2} & \text{otherwise} \end{cases}\right)}{2 \left(-1 + \cot^{2}{\left(1 \right)}\right) \sin^{2}{\left(1 \right)}}$$
// zoo for a mod pi = 0\
// 1 for a mod 2*pi = 0\ || |
|| | || 2/a\ |
2 || 2 | || sin |-| |
2*sin (2)*|< -4 + 4*sin (a) + 4*cos(a) |*|< 1 \2/ |
||--------------------------- otherwise | ||-------- + ------- otherwise |
|| 2 2 | || /a\ sin(a) |
\\2*(1 - cos(a)) + 2*sin (a) / ||2*tan|-| |
\\ \2/ /
----------------------------------------------------------------------------------------------------
sin(4)
$$\frac{2 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{4 \sin^{2}{\left(a \right)} + 4 \cos{\left(a \right)} - 4}{2 \left(- \cos{\left(a \right)} + 1\right)^{2} + 2 \sin^{2}{\left(a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{\sin^{2}{\left(\frac{a}{2} \right)}}{\sin{\left(a \right)}} + \frac{1}{2 \tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) \sin^{2}{\left(2 \right)}}{\sin{\left(4 \right)}}$$
// / pi\ \
|| 0 for |a + --| mod pi = 0| // zoo for a mod pi = 0\
|| \ 2 / | || |
|| | || 2/a\ |
2 / 2 \ || /a pi\ | ||1 + cot |-| |
4*cot (1)*\1 + cot (2)/*|< 2*cot|- + --| |*|< \2/ |
|| \2 4 / | ||----------- otherwise |
||---------------- otherwise | || /a\ |
|| 2/a pi\ | || 2*cot|-| |
||1 + cot |- + --| | \\ \2/ /
\\ \2 4 / /
-----------------------------------------------------------------------------------------------------
2
/ 2 \
\1 + cot (1)/ *cot(2)
$$\frac{4 \left(\cot^{2}{\left(2 \right)} + 1\right) \left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} + 1}{2 \cot{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) \cot^{2}{\left(1 \right)}}{\left(\cot^{2}{\left(1 \right)} + 1\right)^{2} \cot{\left(2 \right)}}$$
// 1 for a mod 2*pi = 0\
|| |
|| 2 | // zoo for a mod pi = 0\
|| sin (a) | || |
||-1 + --------- | || / 2 \ |
|| 4/a\ | || 2/a\ | sin (a) | |
2 || 4*sin |-| | ||sin |-|*|1 + ---------| |
2*sin (2)*|< \2/ |*|< \2/ | 4/a\| |
||-------------- otherwise | || | 4*sin |-|| |
|| 2 | || \ \2// |
|| sin (a) | ||----------------------- otherwise |
||1 + --------- | || sin(a) |
|| 4/a\ | \\ /
|| 4*sin |-| |
\\ \2/ /
--------------------------------------------------------------------------------------------
sin(4)
$$\frac{2 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}}{1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \sin^{2}{\left(\frac{a}{2} \right)}}{\sin{\left(a \right)}} & \text{otherwise} \end{cases}\right) \sin^{2}{\left(2 \right)}}{\sin{\left(4 \right)}}$$
// 1 for a mod 2*pi = 0\ // zoo for a mod pi = 0\
|| | || |
||/ 1 for a mod 2*pi = 0 | ||/ zoo for a mod pi = 0 |
||| | ||| |
||| 2/a\ | ||| 2/a\ |
|<|-1 + cot |-| |*|<|1 + cot |-| |
||< \2/ otherwise | ||< \2/ otherwise |
|||------------ otherwise | |||----------- otherwise |
||| 2/a\ | ||| /a\ |
|||1 + cot |-| | ||| 2*cot|-| |
\\\ \2/ / \\\ \2/ /
------------------------------------------------------------------------------------------------------------
cot(2)
$$\frac{\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} + 1}{2 \cot{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)}{\cot{\left(2 \right)}}$$
// / 3*pi\ \
// 1 for a mod 2*pi = 0\ || 1 for |a + ----| mod 2*pi = 0|
|| | || \ 2 / |
|| 2/a\ | || |
/ 2 \ / 2/ pi\\ ||-1 + cot |-| | || 2/a pi\ |
-\1 + cot (1)/*|-1 + cot |1 + --||*|< \2/ |*|< 1 + tan |- + --| |
\ \ 4 // ||------------ otherwise | || \2 4 / |
|| 2/a\ | ||----------------- otherwise |
||1 + cot |-| | || 2/a pi\ |
\\ \2/ / ||-1 + tan |- + --| |
\\ \2 4 / /
-------------------------------------------------------------------------------------------------------------------------
/ 2/ pi\\ / 2 \
|1 + cot |1 + --||*\-1 + cot (1)/
\ \ 4 //
$$- \frac{\left(-1 + \cot^{2}{\left(\frac{\pi}{4} + 1 \right)}\right) \left(\cot^{2}{\left(1 \right)} + 1\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1} & \text{otherwise} \end{cases}\right)}{\left(-1 + \cot^{2}{\left(1 \right)}\right) \left(\cot^{2}{\left(\frac{\pi}{4} + 1 \right)} + 1\right)}$$
// 1 for a mod 2*pi = 0\
|| |
|| 2/a\ | // zoo for a mod pi = 0\
|| cos |-| | || |
|| \2/ | ||/ 2/a\ \ |
||-1 + ------------ | ||| cos |-| | |
|| 2/a pi\ | ||| \2/ | /a pi\ |
|| cos |- - --| | |||1 + ------------|*cos|- - --| | / pi\
|< \2 2 / |*|<| 2/a pi\| \2 2 / |*cos|2 - --|
||----------------- otherwise | ||| cos |- - --|| | \ 2 /
|| 2/a\ | ||\ \2 2 // |
|| cos |-| | ||------------------------------ otherwise |
|| \2/ | || /a\ |
|| 1 + ------------ | || 2*cos|-| |
|| 2/a pi\ | \\ \2/ /
|| cos |- - --| |
\\ \2 2 / /
--------------------------------------------------------------------------------------------------------
cos(2)
$$\frac{\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{2 \cos{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) \cos{\left(- \frac{\pi}{2} + 2 \right)}}{\cos{\left(2 \right)}}$$
// 1 for a mod 2*pi = 0\
|| |
|| 2/a pi\ | // zoo for a mod pi = 0\
|| sec |- - --| | || |
|| \2 2 / | ||/ 2/a pi\\ |
||-1 + ------------ | ||| sec |- - --|| |
|| 2/a\ | ||| \2 2 /| /a\ |
|| sec |-| | |||1 + ------------|*sec|-| |
|< \2/ |*|<| 2/a\ | \2/ |*sec(2)
||----------------- otherwise | ||| sec |-| | |
|| 2/a pi\ | ||\ \2/ / |
|| sec |- - --| | ||------------------------- otherwise |
|| \2 2 / | || /a pi\ |
|| 1 + ------------ | || 2*sec|- - --| |
|| 2/a\ | \\ \2 2 / /
|| sec |-| |
\\ \2/ /
----------------------------------------------------------------------------------------------
/ pi\
sec|2 - --|
\ 2 /
$$\frac{\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{\left(1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right) \sec{\left(\frac{a}{2} \right)}}{2 \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \sec{\left(2 \right)}}{\sec{\left(- \frac{\pi}{2} + 2 \right)}}$$
// 1 for a mod 2*pi = 0\
|| |
|| 2/a\ | // zoo for a mod pi = 0\
|| csc |-| | || |
|| \2/ | ||/ 2/a\ \ |
||-1 + ------------ | ||| csc |-| | |
|| 2/pi a\ | ||| \2/ | /pi a\ |
|| csc |-- - -| | |||1 + ------------|*csc|-- - -| | / pi\
|< \2 2/ |*|<| 2/pi a\| \2 2/ |*csc|-2 + --|
||----------------- otherwise | ||| csc |-- - -|| | \ 2 /
|| 2/a\ | ||\ \2 2// |
|| csc |-| | ||------------------------------ otherwise |
|| \2/ | || /a\ |
|| 1 + ------------ | || 2*csc|-| |
|| 2/pi a\ | \\ \2/ /
|| csc |-- - -| |
\\ \2 2/ /
---------------------------------------------------------------------------------------------------------
csc(2)
$$\frac{\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{2 \csc{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) \csc{\left(-2 + \frac{\pi}{2} \right)}}{\csc{\left(2 \right)}}$$
Piecewise((1, Mod(a = 2*pi, 0)), ((-1 + csc(a/2)^2/csc(pi/2 - a/2)^2)/(1 + csc(a/2)^2/csc(pi/2 - a/2)^2), True))*Piecewise((±oo, Mod(a = pi, 0)), ((1 + csc(a/2)^2/csc(pi/2 - a/2)^2)*csc(pi/2 - a/2)/(2*csc(a/2)), True))*csc(-2 + pi/2)/csc(2)