Господин Экзамен
Общий знаменатель tan(pi/4+a/2)*((1-sin(a))/cos(a))
Решение
Вы ввели
[src]
/pi a\
tan|-- + -|*(1 - sin(a))
\4 2/
------------------------
cos(a)
$$\frac{\left(- \sin{\left(a \right)} + 1\right) \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\cos{\left(a \right)}}$$
tan(pi/4 + a/2)*(1 - sin(a))/cos(a)
Рациональный знаменатель
[src]
/a pi\ /a pi\
- sin(a)*tan|- + --| + tan|- + --|
\2 4 / \2 4 /
----------------------------------
cos(a)
$$\frac{- \sin{\left(a \right)} \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\cos{\left(a \right)}}$$
/a pi\ /a pi\ tan|- + --| sin(a)*tan|- + --| \2 4 / \2 4 / ----------- - ------------------ cos(a) cos(a)
$$- \frac{\sin{\left(a \right)} \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\cos{\left(a \right)}} + \frac{\tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\cos{\left(a \right)}}$$
tan(a/2 + pi/4)/cos(a) - sin(a)*tan(a/2 + pi/4)/cos(a)
Общий знаменатель
[src]
/a pi\ /a pi\
- sin(a)*tan|- + --| + tan|- + --|
\2 4 / \2 4 /
----------------------------------
cos(a)
$$\frac{- \sin{\left(a \right)} \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\cos{\left(a \right)}}$$
(-sin(a)*tan(a/2 + pi/4) + tan(a/2 + pi/4))/cos(a)
Собрать выражение
[src]
/a pi\
(1 - sin(a))*sec(a)*tan|- + --|
\2 4 /
$$\left(- \sin{\left(a \right)} + 1\right) \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)} \sec{\left(a \right)}$$
(1 - sin(a))*sec(a)*tan(a/2 + pi/4)
Объединение рациональных выражений
[src]
/pi + 2*a\
(1 - sin(a))*tan|--------|
\ 4 /
--------------------------
cos(a)
$$\frac{\left(- \sin{\left(a \right)} + 1\right) \tan{\left(\frac{2 a + \pi}{4} \right)}}{\cos{\left(a \right)}}$$
(1 - sin(a))*tan((pi + 2*a)/4)/cos(a)
Комбинаторика
[src]
/a pi\
-(-1 + sin(a))*tan|- + --|
\2 4 /
---------------------------
cos(a)
$$- \frac{\left(\sin{\left(a \right)} - 1\right) \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\cos{\left(a \right)}}$$
-(-1 + sin(a))*tan(a/2 + pi/4)/cos(a)
Степени
[src]
/ /a pi\ / a pi\\
/ / -I*a I*a\\ | I*|- + --| I*|- - - --||
| I*\- e + e /| | \2 4 / \ 2 4 /|
I*|1 + ------------------|*\- e + e /
\ 2 /
----------------------------------------------------------
/ /a pi\ / a pi\\
/ I*a -I*a\ | I*|- + --| I*|- - - --||
|e e | | \2 4 / \ 2 4 /|
|---- + -----|*\e + e /
\ 2 2 /
$$\frac{i \left(\frac{i \left(e^{i a} - e^{- i a}\right)}{2} + 1\right) \left(e^{i \left(- \frac{a}{2} - \frac{\pi}{4}\right)} - e^{i \left(\frac{a}{2} + \frac{\pi}{4}\right)}\right)}{\left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2}\right) \left(e^{i \left(- \frac{a}{2} - \frac{\pi}{4}\right)} + e^{i \left(\frac{a}{2} + \frac{\pi}{4}\right)}\right)}$$
i*(1 + i*(-exp(-i*a) + exp(i*a))/2)*(-exp(i*(a/2 + pi/4)) + exp(i*(-a/2 - pi/4)))/((exp(i*a)/2 + exp(-i*a)/2)*(exp(i*(a/2 + pi/4)) + exp(i*(-a/2 - pi/4))))
Тригонометрическая часть
[src]
1
$$1$$
1 - sin(a)
----------------------
2
/ /a\\ 2/a\
|-1 + tan|-|| *cos |-|
\ \2// \2/
$$\frac{- \sin{\left(a \right)} + 1}{\left(\tan{\left(\frac{a}{2} \right)} - 1\right)^{2} \cos^{2}{\left(\frac{a}{2} \right)}}$$
2/a\
sec |-|*(1 - sin(a))
\2/
--------------------
2
/ /a\\
|-1 + tan|-||
\ \2//
$$\frac{\left(- \sin{\left(a \right)} + 1\right) \sec^{2}{\left(\frac{a}{2} \right)}}{\left(\tan{\left(\frac{a}{2} \right)} - 1\right)^{2}}$$
2*(1 - sin(a))
---------------------------
2
/ /a\\
(1 + cos(a))*|-1 + tan|-||
\ \2//
$$\frac{2 \cdot \left(- \sin{\left(a \right)} + 1\right)}{\left(\cos{\left(a \right)} + 1\right) \left(\tan{\left(\frac{a}{2} \right)} - 1\right)^{2}}$$
2/a pi\ / / pi\\
2*cos |- - --|*|1 - cos|a - --||
\2 4 / \ \ 2 //
--------------------------------
2
cos (a)
$$\frac{2 \cdot \left(- \cos{\left(a - \frac{\pi}{2} \right)} + 1\right) \cos^{2}{\left(\frac{a}{2} - \frac{\pi}{4} \right)}}{\cos^{2}{\left(a \right)}}$$
/ /a\\
|1 + tan|-||*(-tan(a) + sec(a))
\ \2//
-------------------------------
/a\
1 - tan|-|
\2/
$$\frac{\left(\tan{\left(\frac{a}{2} \right)} + 1\right) \left(- \tan{\left(a \right)} + \sec{\left(a \right)}\right)}{- \tan{\left(\frac{a}{2} \right)} + 1}$$
2 / 1 \
2*sec (a)*|1 - -----------|
| / pi\|
| sec|a - --||
\ \ 2 //
---------------------------
2/a pi\
sec |- - --|
\2 4 /
$$\frac{2 \cdot \left(1 - \frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}}\right) \sec^{2}{\left(a \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{4} \right)}}$$
/ /a\\ / 1 \
|1 + tan|-||*|------ - tan(a)|
\ \2// \cos(a) /
------------------------------
/a\
1 - tan|-|
\2/
$$\frac{\left(\tan{\left(\frac{a}{2} \right)} + 1\right) \left(- \tan{\left(a \right)} + \frac{1}{\cos{\left(a \right)}}\right)}{- \tan{\left(\frac{a}{2} \right)} + 1}$$
/a pi\
(1 - sin(a))*sin|- + --|
\2 4 /
------------------------
/a pi\
cos(a)*cos|- + --|
\2 4 /
$$\frac{\left(- \sin{\left(a \right)} + 1\right) \sin{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\cos{\left(a \right)} \cos{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
2/a pi\
2*sin |- + --|*(1 - sin(a))
\2 4 /
---------------------------
/ pi\
cos(a)*sin|a + --|
\ 2 /
$$\frac{2 \cdot \left(- \sin{\left(a \right)} + 1\right) \sin^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\sin{\left(a + \frac{\pi}{2} \right)} \cos{\left(a \right)}}$$
/ 1 \ /a pi\
|1 - ------|*sec(a)*sec|- + --|
\ csc(a)/ \2 4 /
-------------------------------
/a pi\
csc|- + --|
\2 4 /
$$\frac{\left(1 - \frac{1}{\csc{\left(a \right)}}\right) \sec{\left(a \right)} \sec{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\csc{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
1 - sin(a)
------------------------------
2
/ 2/a\\
| 2*sin |-||
| \2/| 2/pi a\
|-1 + ---------| *sin |-- + -|
\ sin(a) / \2 2/
$$\frac{- \sin{\left(a \right)} + 1}{\left(\frac{2 \sin^{2}{\left(\frac{a}{2} \right)}}{\sin{\left(a \right)}} - 1\right)^{2} \sin^{2}{\left(\frac{a}{2} + \frac{\pi}{2} \right)}}$$
/ 1 \ /pi \
2*|1 - ------|*csc|-- - a|*sec(a)
\ csc(a)/ \2 /
---------------------------------
2/a pi\
csc |- + --|
\2 4 /
$$\frac{2 \cdot \left(1 - \frac{1}{\csc{\left(a \right)}}\right) \csc{\left(- a + \frac{\pi}{2} \right)} \sec{\left(a \right)}}{\csc^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
/ / pi\\ /a pi\
|1 - cos|a - --||*cos|- - --|
\ \ 2 // \2 4 /
-----------------------------
/a pi\
cos(a)*cos|- + --|
\2 4 /
$$\frac{\left(- \cos{\left(a - \frac{\pi}{2} \right)} + 1\right) \cos{\left(\frac{a}{2} - \frac{\pi}{4} \right)}}{\cos{\left(a \right)} \cos{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
2*(1 - sin(a))
------------------------------------
2
/ 1 1 \
(1 + cos(a))*|-1 + ------ - ------|
\ sin(a) tan(a)/
$$\frac{2 \cdot \left(- \sin{\left(a \right)} + 1\right)}{\left(\cos{\left(a \right)} + 1\right) \left(-1 - \frac{1}{\tan{\left(a \right)}} + \frac{1}{\sin{\left(a \right)}}\right)^{2}}$$
1 - sin(a)
-------------------------------------
2 2
/ 2/a\\ / /a\\ 4/a\
|1 - tan |-|| *|-1 + tan|-|| *cos |-|
\ \4// \ \2// \4/
$$\frac{- \sin{\left(a \right)} + 1}{\left(- \tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2} \left(\tan{\left(\frac{a}{2} \right)} - 1\right)^{2} \cos^{4}{\left(\frac{a}{4} \right)}}$$
/a pi\
(1 - sin(a))*sin|- + --|
\2 4 /
-------------------------
/ pi\ /a 3*pi\
sin|a + --|*sin|- + ----|
\ 2 / \2 4 /
$$\frac{\left(- \sin{\left(a \right)} + 1\right) \sin{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\sin{\left(\frac{a}{2} + \frac{3 \pi}{4} \right)} \sin{\left(a + \frac{\pi}{2} \right)}}$$
/ 1 \ /a pi\
|1 - -----------|*sec(a)*sec|- + --|
| / pi\| \2 4 /
| sec|a - --||
\ \ 2 //
------------------------------------
/a pi\
sec|- - --|
\2 4 /
$$\frac{\left(1 - \frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}}\right) \sec{\left(a \right)} \sec{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\sec{\left(\frac{a}{2} - \frac{\pi}{4} \right)}}$$
/ 1 \ /a pi\
|1 - -----------|*sec(a)*sec|- + --|
| /pi \| \2 4 /
| sec|-- - a||
\ \2 //
------------------------------------
/ a pi\
sec|- - + --|
\ 2 4 /
$$\frac{\left(1 - \frac{1}{\sec{\left(- a + \frac{\pi}{2} \right)}}\right) \sec{\left(a \right)} \sec{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\sec{\left(- \frac{a}{2} + \frac{\pi}{4} \right)}}$$
/ 1 \ /pi \ / a pi\
|1 - ------|*csc|-- - a|*csc|- - + --|
\ csc(a)/ \2 / \ 2 4 /
--------------------------------------
/a pi\
csc|- + --|
\2 4 /
$$\frac{\left(1 - \frac{1}{\csc{\left(a \right)}}\right) \csc{\left(- a + \frac{\pi}{2} \right)} \csc{\left(- \frac{a}{2} + \frac{\pi}{4} \right)}}{\csc{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
/ pi\
1 - cos|a - --|
\ 2 /
---------------------------
2
/ /a pi\\
| cos|- - --||
| \2 2 /| 2/a\
|-1 + -----------| *cos |-|
| /a\ | \2/
| cos|-| |
\ \2/ /
$$\frac{- \cos{\left(a - \frac{\pi}{2} \right)} + 1}{\left(-1 + \frac{\cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos{\left(\frac{a}{2} \right)}}\right)^{2} \cos^{2}{\left(\frac{a}{2} \right)}}$$
/ 1 \ /pi \ / a pi\
|1 - -----------|*csc|-- - a|*csc|- - + --|
\ csc(pi - a)/ \2 / \ 2 4 /
-------------------------------------------
/ a 3*pi\
csc|- - + ----|
\ 2 4 /
$$\frac{\left(1 - \frac{1}{\csc{\left(- a + \pi \right)}}\right) \csc{\left(- a + \frac{\pi}{2} \right)} \csc{\left(- \frac{a}{2} + \frac{\pi}{4} \right)}}{\csc{\left(- \frac{a}{2} + \frac{3 \pi}{4} \right)}}$$
2/a\ / 1 \
sec |-|*|1 - -----------|
\2/ | / pi\|
| sec|a - --||
\ \ 2 //
-------------------------
2
/ /a\ \
| sec|-| |
| \2/ |
|-1 + -----------|
| /a pi\|
| sec|- - --||
\ \2 2 //
$$\frac{\left(1 - \frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}}\right) \sec^{2}{\left(\frac{a}{2} \right)}}{\left(\frac{\sec{\left(\frac{a}{2} \right)}}{\sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - 1\right)^{2}}$$
2/pi a\ / 1 \
csc |-- - -|*|1 - ------|
\2 2/ \ csc(a)/
-------------------------
2
/ /pi a\\
| csc|-- - -||
| \2 2/|
|-1 + -----------|
| /a\ |
| csc|-| |
\ \2/ /
$$\frac{\left(1 - \frac{1}{\csc{\left(a \right)}}\right) \csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\left(-1 + \frac{\csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc{\left(\frac{a}{2} \right)}}\right)^{2}}$$
/ 2/a\\
| 1 + tan |-||
/ /a\\ | \2/|
|1 + tan|-||*|-tan(a) + -----------|
\ \2// | 2/a\|
| 1 - tan |-||
\ \2//
------------------------------------
/a\
1 - tan|-|
\2/
$$\frac{\left(\tan{\left(\frac{a}{2} \right)} + 1\right) \left(- \tan{\left(a \right)} + \frac{\tan^{2}{\left(\frac{a}{2} \right)} + 1}{- \tan^{2}{\left(\frac{a}{2} \right)} + 1}\right)}{- \tan{\left(\frac{a}{2} \right)} + 1}$$
/ 1 \ / 1 1 \
|------ - tan(a)|*|1 + ------ - ------|
\cos(a) / \ sin(a) tan(a)/
---------------------------------------
1 1
1 + ------ - ------
tan(a) sin(a)
$$\frac{\left(- \tan{\left(a \right)} + \frac{1}{\cos{\left(a \right)}}\right) \left(1 - \frac{1}{\tan{\left(a \right)}} + \frac{1}{\sin{\left(a \right)}}\right)}{1 + \frac{1}{\tan{\left(a \right)}} - \frac{1}{\sin{\left(a \right)}}}$$
/ /a\\ |1 + tan|-||*(-1 + sin(a)) \ \2// -------------------------------- / /a\\ / 2/a\ 2/a\\ |1 - tan|-||*|sin |-| - cos |-|| \ \2// \ \2/ \2//
$$\frac{\left(\sin{\left(a \right)} - 1\right) \left(\tan{\left(\frac{a}{2} \right)} + 1\right)}{\left(- \tan{\left(\frac{a}{2} \right)} + 1\right) \left(\sin^{2}{\left(\frac{a}{2} \right)} - \cos^{2}{\left(\frac{a}{2} \right)}\right)}$$
/ /a\\
| tan|-||
/ /a\\ | 1 /a\ \2/|
|1 + tan|-||*|------ - tan|-| - ------|
\ \2// \cos(a) \2/ cos(a)/
---------------------------------------
/a\
1 - tan|-|
\2/
$$\frac{\left(\tan{\left(\frac{a}{2} \right)} + 1\right) \left(- \tan{\left(\frac{a}{2} \right)} - \frac{\tan{\left(\frac{a}{2} \right)}}{\cos{\left(a \right)}} + \frac{1}{\cos{\left(a \right)}}\right)}{- \tan{\left(\frac{a}{2} \right)} + 1}$$
/ /a\ \
| 2*tan|-| |
/ 2/a\\ | \2/ | /a pi\
|1 + tan |-||*|1 - -----------|*tan|- + --|
\ \2// | 2/a\| \2 4 /
| 1 + tan |-||
\ \2//
-------------------------------------------
2/a\
1 - tan |-|
\2/
$$\frac{\left(1 - \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{- \tan^{2}{\left(\frac{a}{2} \right)} + 1}$$
/ // 1 for a mod 2*pi = 0\\
/ /a\\ | || ||
|1 + tan|-||*|-tan(a) + |< 1 ||
\ \2// | ||------ otherwise ||
\ \\cos(a) //
------------------------------------------------------
/a\
1 - tan|-|
\2/
$$\frac{\left(\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\cos{\left(a \right)}} & \text{otherwise} \end{cases}\right) - \tan{\left(a \right)}\right) \left(\tan{\left(\frac{a}{2} \right)} + 1\right)}{- \tan{\left(\frac{a}{2} \right)} + 1}$$
/a pi\
2*(1 - sin(a))*tan|- + --|
\4 8 /
----------------------------------------
/ 2/a\\ / 2/a pi\\ 2/a\
|1 - tan |-||*|1 - tan |- + --||*cos |-|
\ \2// \ \4 8 // \2/
$$\frac{2 \cdot \left(- \sin{\left(a \right)} + 1\right) \tan{\left(\frac{a}{4} + \frac{\pi}{8} \right)}}{\left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(- \tan^{2}{\left(\frac{a}{4} + \frac{\pi}{8} \right)} + 1\right) \cos^{2}{\left(\frac{a}{2} \right)}}$$
/ /a\ \
2 | 2*tan|-| |
/ 2/a\\ | \2/ |
|1 + tan |-|| *|1 - -----------|
\ \4// | 2/a\|
| 1 + tan |-||
\ \2//
--------------------------------
2 2
/ 2/a\\ / /a\\
|1 - tan |-|| *|-1 + tan|-||
\ \4// \ \2//
$$\frac{\left(1 - \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2}}{\left(- \tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2} \left(\tan{\left(\frac{a}{2} \right)} - 1\right)^{2}}$$
/ 2/a\\
| 2*sin |-|| / 2 \
| \2/| | 1 2*sin (a)|
|1 + ---------|*|----------- - ---------|
\ sin(a) / | / pi\ sin(2*a)|
|sin|a + --| |
\ \ 2 / /
-----------------------------------------
2/a\
2*sin |-|
\2/
1 - ---------
sin(a)
$$\frac{\left(\frac{2 \sin^{2}{\left(\frac{a}{2} \right)}}{\sin{\left(a \right)}} + 1\right) \left(- \frac{2 \sin^{2}{\left(a \right)}}{\sin{\left(2 a \right)}} + \frac{1}{\sin{\left(a + \frac{\pi}{2} \right)}}\right)}{- \frac{2 \sin^{2}{\left(\frac{a}{2} \right)}}{\sin{\left(a \right)}} + 1}$$
/ 1 \ / 2 \
|1 + -------|*|1 - --------------------|
| 2/a\| | / 1 \ /a\|
| cot |-|| | |1 + -------|*cot|-||
\ \2// | | 2/a\| \2/|
| | cot |-|| |
\ \ \2// /
----------------------------------------
/ 1 \ /a pi\
|1 - -------|*cot|- + --|
| 2/a\| \2 4 /
| cot |-||
\ \2//
$$\frac{\left(1 - \frac{2}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \cot{\left(\frac{a}{2} \right)}}\right) \left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right)}{\left(1 - \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
// 1 for a mod 2*pi = 0\
/ // 0 for a mod pi = 0\\ || | /a pi\
|1 - |< ||*|< 1 |*tan|- + --|
\ \\sin(a) otherwise // ||------ otherwise | \2 4 /
\\cos(a) /
$$\left(\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\cos{\left(a \right)}} & \text{otherwise} \end{cases}\right) \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}$$
/ /a\ \
| sec|-| |
| \2/ | / sec(a) \
|1 + -----------|*|- ----------- + sec(a)|
| /a pi\| | / pi\ |
| sec|- - --|| | sec|a - --| |
\ \2 2 // \ \ 2 / /
------------------------------------------
/a\
sec|-|
\2/
1 - -----------
/a pi\
sec|- - --|
\2 2 /
$$\frac{\left(\frac{\sec{\left(\frac{a}{2} \right)}}{\sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \left(\sec{\left(a \right)} - \frac{\sec{\left(a \right)}}{\sec{\left(a - \frac{\pi}{2} \right)}}\right)}{- \frac{\sec{\left(\frac{a}{2} \right)}}{\sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1}$$
/ /a pi\\ / / pi\\
| cos|- - --|| | cos|a - --||
| \2 2 /| | 1 \ 2 /|
|1 + -----------|*|------ - -----------|
| /a\ | \cos(a) cos(a) /
| cos|-| |
\ \2/ /
----------------------------------------
/a pi\
cos|- - --|
\2 2 /
1 - -----------
/a\
cos|-|
\2/
$$\frac{\left(1 + \frac{\cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos{\left(\frac{a}{2} \right)}}\right) \left(- \frac{\cos{\left(a - \frac{\pi}{2} \right)}}{\cos{\left(a \right)}} + \frac{1}{\cos{\left(a \right)}}\right)}{1 - \frac{\cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos{\left(\frac{a}{2} \right)}}}$$
// 1 for a mod 2*pi = 0\
/ // 0 for a mod pi = 0\\ || |
|1 - |< ||*(1 + sin(a))*|< 1 |
\ \\sin(a) otherwise // ||------ otherwise |
\\cos(a) /
----------------------------------------------------------------------------
cos(a)
$$\frac{\left(\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1\right) \left(\sin{\left(a \right)} + 1\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\cos{\left(a \right)}} & \text{otherwise} \end{cases}\right)}{\cos{\left(a \right)}}$$
/ /pi a\\ / /pi \ \
| csc|-- - -|| | csc|-- - a| |
| \2 2/| | \2 / /pi \|
|1 + -----------|*|- ----------- + csc|-- - a||
| /a\ | \ csc(a) \2 //
| csc|-| |
\ \2/ /
-----------------------------------------------
/pi a\
csc|-- - -|
\2 2/
1 - -----------
/a\
csc|-|
\2/
$$\frac{\left(1 + \frac{\csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc{\left(\frac{a}{2} \right)}}\right) \left(\csc{\left(- a + \frac{\pi}{2} \right)} - \frac{\csc{\left(- a + \frac{\pi}{2} \right)}}{\csc{\left(a \right)}}\right)}{1 - \frac{\csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc{\left(\frac{a}{2} \right)}}}$$
// a \
|| 1 for - mod 2*pi = 0|
/ // 0 for a mod pi = 0\\ || 2 |
|1 - |< ||*|< |
\ \\sin(a) otherwise // || 2 |
||---------- otherwise |
\\1 + cos(a) /
-------------------------------------------------------------------
2
/ /a\\
|-1 + tan|-||
\ \2//
$$\frac{\left(\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1\right) \left(\begin{cases} 1 & \text{for}\: \frac{a}{2} \bmod 2 \pi = 0 \\\frac{2}{\cos{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right)}{\left(\tan{\left(\frac{a}{2} \right)} - 1\right)^{2}}$$
/ /a\ \
| 2*tan|-| |
/ 2/a\\ | \2/ | /a pi\
2*|1 + tan |-||*|1 - -----------|*tan|- + --|
\ \2// | 2/a\| \4 8 /
| 1 + tan |-||
\ \2//
---------------------------------------------
/ 2/a\\ / 2/a pi\\
|1 - tan |-||*|1 - tan |- + --||
\ \2// \ \4 8 //
$$\frac{2 \cdot \left(1 - \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \tan{\left(\frac{a}{4} + \frac{\pi}{8} \right)}}{\left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(- \tan^{2}{\left(\frac{a}{4} + \frac{\pi}{8} \right)} + 1\right)}$$
/ // 1 for a mod 2*pi = 0\\
| || ||
| || 2/a\ ||
/ 1 \ | 1 ||1 + cot |-| ||
|1 + ------|*|- ------ + |< \2/ ||
| /a\| | cot(a) ||------------ otherwise ||
| cot|-|| | || 2/a\ ||
\ \2// | ||-1 + cot |-| ||
\ \\ \2/ //
-------------------------------------------------------------
1
1 - ------
/a\
cot|-|
\2/
$$\frac{\left(1 + \frac{1}{\cot{\left(\frac{a}{2} \right)}}\right) \left(\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} + 1}{\cot^{2}{\left(\frac{a}{2} \right)} - 1} & \text{otherwise} \end{cases}\right) - \frac{1}{\cot{\left(a \right)}}\right)}{1 - \frac{1}{\cot{\left(\frac{a}{2} \right)}}}$$
/ // 0 for a mod pi = 0\\ | || || // 1 for a mod 2*pi = 0\ | ||1 - cos(a) || || | /a pi\ |1 - |<---------- otherwise ||*|< 1 |*tan|- + --| | || /a\ || ||------ otherwise | \2 4 / | || tan|-| || \\cos(a) / \ \\ \2/ //
$$\left(\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{- \cos{\left(a \right)} + 1}{\tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\cos{\left(a \right)}} & \text{otherwise} \end{cases}\right) \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}$$
// 1 for a mod 2*pi = 0\
|| |
2/a pi\ / // 0 for a mod pi = 0\\ || 1 |
2*sin |- + --|*|1 - |< ||*|<----------- otherwise |
\2 4 / \ \\sin(a) otherwise // || / pi\ |
||sin|a + --| |
\\ \ 2 / /
-----------------------------------------------------------------------------------
cos(a)
$$\frac{2 \cdot \left(\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\sin{\left(a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \sin^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\cos{\left(a \right)}}$$
/ // 0 for a mod pi = 0\\
| || ||
| || 1 || // 1 for a mod 2*pi = 0\ /a pi\
|1 - |<----------- otherwise ||*|< |*sec|- + --|
| || / pi\ || \\sec(a) otherwise / \2 4 /
| ||sec|a - --| ||
\ \\ \ 2 / //
--------------------------------------------------------------------------------
/a pi\
sec|- - --|
\2 4 /
$$\frac{\left(\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\sec{\left(a \right)} & \text{otherwise} \end{cases}\right) \sec{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\sec{\left(\frac{a}{2} - \frac{\pi}{4} \right)}}$$
/ // 0 for a mod pi = 0\\ // 1 for a mod 2*pi = 0\
| || || || | /a pi\
|1 - |< / pi\ ||*|< 1 |*cos|- - --|
| ||cos|a - --| otherwise || ||------ otherwise | \2 4 /
\ \\ \ 2 / // \\cos(a) /
--------------------------------------------------------------------------------
/a pi\
cos|- + --|
\2 4 /
$$\frac{\left(\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + 1\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\cos{\left(a \right)}} & \text{otherwise} \end{cases}\right) \cos{\left(\frac{a}{2} - \frac{\pi}{4} \right)}}{\cos{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
/ // 0 for a mod pi = 0\\ // 1 for a mod 2*pi = 0\
| || || || | / a pi\
|1 - |< 1 ||*|< /pi \ |*csc|- - + --|
| ||------ otherwise || ||csc|-- - a| otherwise | \ 2 4 /
\ \\csc(a) // \\ \2 / /
----------------------------------------------------------------------------------
/a pi\
csc|- + --|
\2 4 /
$$\frac{\left(\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\csc{\left(a \right)}} & \text{otherwise} \end{cases}\right) + 1\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\csc{\left(- a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \csc{\left(- \frac{a}{2} + \frac{\pi}{4} \right)}}{\csc{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
2/a pi\ / 2/a pi\\ / 2/a 3*pi\\ /a pi\
sin |- + --|*|1 + tan |- + --||*|1 + tan |- + ----||*(1 - sin(a))*cot|- + --|
\4 8 / \ \2 4 // \ \4 8 // \4 8 /
-----------------------------------------------------------------------------
/a pi\ /a 3*pi\
2*tan|- + --|*tan|- + ----|
\2 4 / \4 8 /
$$\frac{\left(- \sin{\left(a \right)} + 1\right) \left(\tan^{2}{\left(\frac{a}{4} + \frac{3 \pi}{8} \right)} + 1\right) \left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \sin^{2}{\left(\frac{a}{4} + \frac{\pi}{8} \right)} \cot{\left(\frac{a}{4} + \frac{\pi}{8} \right)}}{2 \tan{\left(\frac{a}{4} + \frac{3 \pi}{8} \right)} \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
2/a pi\ / 2/a pi\\
4*tan |- + --|*|1 + tan |- + --||*(1 - sin(a))
\4 8 / \ \2 4 //
-----------------------------------------------------
2
/ 2/a pi\\ / 2/a\\ 2/a\ /a pi\
|1 + tan |- + --|| *|1 - tan |-||*cos |-|*tan|- + --|
\ \4 8 // \ \2// \2/ \2 4 /
$$\frac{4 \cdot \left(- \sin{\left(a \right)} + 1\right) \left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \tan^{2}{\left(\frac{a}{4} + \frac{\pi}{8} \right)}}{\left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{a}{4} + \frac{\pi}{8} \right)} + 1\right)^{2} \cos^{2}{\left(\frac{a}{2} \right)} \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
/ 2/a 3*pi\\
(1 - sin(a))*|-1 + tan |- + ----||
\ \4 8 //
----------------------------------------------------------------------------
/ 2/a 3*pi\\ / 2/a\\ / 2/a pi\\ 2/a\ 2/a pi\
|1 + tan |- + ----||*|-1 + cot |-||*|-1 + cot |- + --||*sin |-|*sin |- + --|
\ \4 8 // \ \2// \ \4 8 // \2/ \4 8 /
$$\frac{\left(- \sin{\left(a \right)} + 1\right) \left(\tan^{2}{\left(\frac{a}{4} + \frac{3 \pi}{8} \right)} - 1\right)}{\left(\tan^{2}{\left(\frac{a}{4} + \frac{3 \pi}{8} \right)} + 1\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right) \left(\cot^{2}{\left(\frac{a}{4} + \frac{\pi}{8} \right)} - 1\right) \sin^{2}{\left(\frac{a}{2} \right)} \sin^{2}{\left(\frac{a}{4} + \frac{\pi}{8} \right)}}$$
/ 4/a\\
| 4*sin |-||
2 | \2/|
(-1 + cos(a) + sin(a)) *|1 + ---------|*(1 + sin(a))
| 2 |
\ sin (a) /
-----------------------------------------------------
/ 4/a\\
| 4*sin |-||
| \2/| /1 2 cos(2*a)\
|1 - ---------|*|- + (1 - cos(a)) - --------|*cos(a)
| 2 | \2 2 /
\ sin (a) /
$$\frac{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right) \left(\sin{\left(a \right)} + 1\right) \left(\sin{\left(a \right)} + \cos{\left(a \right)} - 1\right)^{2}}{\left(- \frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right) \left(\left(- \cos{\left(a \right)} + 1\right)^{2} - \frac{\cos{\left(2 a \right)}}{2} + \frac{1}{2}\right) \cos{\left(a \right)}}$$
/ // 0 for a mod pi = 0\\ // 1 for a mod 2*pi = 0\ | || || || | | || /a\ || || 2/a\ | | || 2*tan|-| || ||1 + tan |-| | /a pi\ |1 - |< \2/ ||*|< \2/ |*tan|- + --| | ||----------- otherwise || ||----------- otherwise | \2 4 / | || 2/a\ || || 2/a\ | | ||1 + tan |-| || ||1 - tan |-| | \ \\ \2/ // \\ \2/ /
$$\left(\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{a}{2} \right)} + 1}{- \tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}$$
/ // 0 for a mod pi = 0\\ // 1 for a mod 2*pi = 0\
| || || || |
| || /a\ || || 2/a\ |
| || 2*cot|-| || ||1 + cot |-| |
|1 - |< \2/ ||*|< \2/ |
| ||----------- otherwise || ||------------ otherwise |
| || 2/a\ || || 2/a\ |
| ||1 + cot |-| || ||-1 + cot |-| |
\ \\ \2/ // \\ \2/ /
--------------------------------------------------------------------------
/a pi\
cot|- + --|
\2 4 /
$$\frac{\left(\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} + 1}{\cot^{2}{\left(\frac{a}{2} \right)} - 1} & \text{otherwise} \end{cases}\right)}{\cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
/ 4/a\\ / 2/a\ \
| 4*sin |-|| | 4*sin |-| |
2/a pi\ | \2/| | \2/ |
2*sin |- + --|*|1 + ---------|*|1 - ----------------------|
\2 4 / | 2 | | / 4/a\\ |
\ sin (a) / | | 4*sin |-|| |
| | \2/| |
| |1 + ---------|*sin(a)|
| | 2 | |
\ \ sin (a) / /
-----------------------------------------------------------
/ 4/a\\
| 4*sin |-||
| \2/|
|1 - ---------|*cos(a)
| 2 |
\ sin (a) /
$$\frac{2 \cdot \left(1 - \frac{4 \sin^{2}{\left(\frac{a}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right) \sin{\left(a \right)}}\right) \left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right) \sin^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(- \frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right) \cos{\left(a \right)}}$$
/ /a\ \
| 2*cot|-| |
/ 2/a pi\\ / 2/a 3*pi\\ | \2/ | /a pi\
|1 + tan |- + --||*|1 + tan |- + ----||*|1 - -----------|*cot|- + --|
\ \2 4 // \ \4 8 // | 2/a\| \4 8 /
| 1 + cot |-||
\ \2//
---------------------------------------------------------------------
/ 2/a pi\\ /a pi\ /a 3*pi\
2*|1 + cot |- + --||*tan|- + --|*tan|- + ----|
\ \4 8 // \2 4 / \4 8 /
$$\frac{\left(1 - \frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{a}{4} + \frac{3 \pi}{8} \right)} + 1\right) \left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \cot{\left(\frac{a}{4} + \frac{\pi}{8} \right)}}{2 \left(\cot^{2}{\left(\frac{a}{4} + \frac{\pi}{8} \right)} + 1\right) \tan{\left(\frac{a}{4} + \frac{3 \pi}{8} \right)} \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
/ /a\ \
| 2*tan|-| |
2/a pi\ / 2/a\\ / 2/a pi\\ | \2/ |
4*tan |- + --|*|1 + tan |-||*|1 + tan |- + --||*|1 - -----------|
\4 8 / \ \2// \ \2 4 // | 2/a\|
| 1 + tan |-||
\ \2//
-----------------------------------------------------------------
2
/ 2/a pi\\ / 2/a\\ /a pi\
|1 + tan |- + --|| *|1 - tan |-||*tan|- + --|
\ \4 8 // \ \2// \2 4 /
$$\frac{4 \cdot \left(1 - \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \tan^{2}{\left(\frac{a}{4} + \frac{\pi}{8} \right)}}{\left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{a}{4} + \frac{\pi}{8} \right)} + 1\right)^{2} \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
// 1 for a mod 2*pi = 0\
|| |
/ // 0 for a mod pi = 0\\ || 1 |
| || || ||1 + ------- |
| || 2 || || 2/a\ |
| ||-------------------- otherwise || || tan |-| | /a pi\
|1 - |
$$\left(\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right) \tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}}{-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right) \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}$$
// 1 for a mod 2*pi = 0\
/ // 0 for a mod pi = 0\\ || |
| || || ||/ 1 for a mod 2*pi = 0 | /a pi\
|1 - |
$$\left(\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + 1\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\cos{\left(a \right)}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}$$
// a \
|| 1 for - mod 2*pi = 0|
/ // 0 for a mod pi = 0\\ || 2 |
| || || || |
| || /a\ || || 2 |
| || 2*cot|-| || || / 2/a\\ |
|1 - |< \2/ ||*|< |1 + cot |-|| |
| ||----------- otherwise || || \ \4// |
| || 2/a\ || ||--------------- otherwise |
| ||1 + cot |-| || || 2 |
\ \\ \2/ // ||/ 2/a\\ |
|||-1 + cot |-|| |
\\\ \4// /
-----------------------------------------------------------------------------
2
/ 1 \
|-1 + ------|
| /a\|
| cot|-||
\ \2//
$$\frac{\left(\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1\right) \left(\begin{cases} 1 & \text{for}\: \frac{a}{2} \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{2}} & \text{otherwise} \end{cases}\right)}{\left(-1 + \frac{1}{\cot{\left(\frac{a}{2} \right)}}\right)^{2}}$$
/ / 2/a pi\\ \
| |1 - cot |- + --||*(1 + sin(a))|
2/a 3*pi\ / 2/a pi\\ / 2/a 3*pi\\ | \ \2 4 // |
sin |- + ----|*|1 + tan |- + --||*|1 - cot |- + ----||*|1 - -------------------------------|
\4 8 / \ \4 8 // \ \4 8 // \ 2 /
--------------------------------------------------------------------------------------------
/ 2/a\\ / 2/a pi\\ 2/a\
|1 - tan |-||*|1 - tan |- + --||*cos |-|
\ \2// \ \4 8 // \2/
$$\frac{\left(- \cot^{2}{\left(\frac{a}{4} + \frac{3 \pi}{8} \right)} + 1\right) \left(- \frac{\left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(a \right)} + 1\right)}{2} + 1\right) \left(\tan^{2}{\left(\frac{a}{4} + \frac{\pi}{8} \right)} + 1\right) \sin^{2}{\left(\frac{a}{4} + \frac{3 \pi}{8} \right)}}{\left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(- \tan^{2}{\left(\frac{a}{4} + \frac{\pi}{8} \right)} + 1\right) \cos^{2}{\left(\frac{a}{2} \right)}}$$
/ 2/a pi\\
| -1 + tan |- + --||
/ 2/a\\ / 2/a pi\\ | \2 4 /| / 2/a 3*pi\\
|1 + cot |-||*|1 + cot |- + --||*|1 - -----------------|*|-1 + tan |- + ----||
\ \2// \ \4 8 // | 2/a pi\| \ \4 8 //
| 1 + tan |- + --||
\ \2 4 //
------------------------------------------------------------------------------
/ 2/a 3*pi\\ / 2/a\\ / 2/a pi\\
|1 + tan |- + ----||*|-1 + cot |-||*|-1 + cot |- + --||
\ \4 8 // \ \2// \ \4 8 //
$$\frac{\left(- \frac{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{a}{4} + \frac{3 \pi}{8} \right)} - 1\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{a}{4} + \frac{\pi}{8} \right)} + 1\right)}{\left(\tan^{2}{\left(\frac{a}{4} + \frac{3 \pi}{8} \right)} + 1\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right) \left(\cot^{2}{\left(\frac{a}{4} + \frac{\pi}{8} \right)} - 1\right)}$$
/ 2/a pi\\
| 1 - cot |- + --||
/ 2/a\\ / 2/a pi\\ / 2/a 3*pi\\ | \2 4 /|
|1 + tan |-||*|1 + tan |- + --||*|1 - cot |- + ----||*|1 - ----------------|
\ \2// \ \4 8 // \ \4 8 // | 2/a pi\|
| 1 + cot |- + --||
\ \2 4 //
----------------------------------------------------------------------------
/ 2/a 3*pi\\ / 2/a\\ / 2/a pi\\
|1 + cot |- + ----||*|1 - tan |-||*|1 - tan |- + --||
\ \4 8 // \ \2// \ \4 8 //
$$\frac{\left(- \cot^{2}{\left(\frac{a}{4} + \frac{3 \pi}{8} \right)} + 1\right) \left(- \frac{- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{a}{4} + \frac{\pi}{8} \right)} + 1\right)}{\left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(- \tan^{2}{\left(\frac{a}{4} + \frac{\pi}{8} \right)} + 1\right) \left(\cot^{2}{\left(\frac{a}{4} + \frac{3 \pi}{8} \right)} + 1\right)}$$
/ 2/a pi\\ / /a pi\ \
| cos |- - --|| | 2*cos|- - --| |
| \2 2 /| | \2 2 / | /a pi\
|1 + ------------|*|1 - -------------------------|*cos|- - --|
| 2/a\ | | / 2/a pi\\ | \2 4 /
| cos |-| | | | cos |- - --|| |
\ \2/ / | | \2 2 /| /a\|
| |1 + ------------|*cos|-||
| | 2/a\ | \2/|
| | cos |-| | |
\ \ \2/ / /
--------------------------------------------------------------
/ 2/a pi\\
| cos |- - --||
| \2 2 /| /a pi\
|1 - ------------|*cos|- + --|
| 2/a\ | \2 4 /
| cos |-| |
\ \2/ /
$$\frac{\left(1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right) \left(1 - \frac{2 \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right) \cos{\left(\frac{a}{2} \right)}}\right) \cos{\left(\frac{a}{2} - \frac{\pi}{4} \right)}}{\left(1 - \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right) \cos{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
/ 2/a\ \ / /a\ \
| sec |-| | | 2*sec|-| |
| \2/ | | \2/ | /a pi\
|1 + ------------|*|1 - ------------------------------|*sec|- + --|
| 2/a pi\| | / 2/a\ \ | \2 4 /
| sec |- - --|| | | sec |-| | |
\ \2 2 // | | \2/ | /a pi\|
| |1 + ------------|*sec|- - --||
| | 2/a pi\| \2 2 /|
| | sec |- - --|| |
\ \ \2 2 // /
-------------------------------------------------------------------
/ 2/a\ \
| sec |-| |
| \2/ | /a pi\
|1 - ------------|*sec|- - --|
| 2/a pi\| \2 4 /
| sec |- - --||
\ \2 2 //
$$\frac{\left(1 - \frac{2 \sec{\left(\frac{a}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}\right) \left(\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(- \frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{a}{2} - \frac{\pi}{4} \right)}}$$
/ 2/pi a\\ / /pi a\ \
| csc |-- - -|| | 2*csc|-- - -| |
| \2 2/| | \2 2/ | / a pi\
|1 + ------------|*|1 - -------------------------|*csc|- - + --|
| 2/a\ | | / 2/pi a\\ | \ 2 4 /
| csc |-| | | | csc |-- - -|| |
\ \2/ / | | \2 2/| /a\|
| |1 + ------------|*csc|-||
| | 2/a\ | \2/|
| | csc |-| | |
\ \ \2/ / /
----------------------------------------------------------------
/ 2/pi a\\
| csc |-- - -||
| \2 2/| /a pi\
|1 - ------------|*csc|- + --|
| 2/a\ | \2 4 /
| csc |-| |
\ \2/ /
$$\frac{\left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right) \left(1 - \frac{2 \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right) \csc{\left(\frac{a}{2} \right)}}\right) \csc{\left(- \frac{a}{2} + \frac{\pi}{4} \right)}}{\left(1 - \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right) \csc{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
/ // 0 for a mod pi = 0\\
| || ||
| || 2*sin(a) || // 1 for a mod 2*pi = 0\
| ||---------------------------- otherwise || || |
| || / 2 \ || || 2 2 |
|1 - |< | sin (a) | ||*(1 + 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// ||
\ \\ //
---------------------------------------------------------------------------------------------------------------------
cos(a)
$$\frac{\left(\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \sin{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \left(- \cos{\left(a \right)} + 1\right)} & \text{otherwise} \end{cases}\right) + 1\right) \left(\sin{\left(a \right)} + 1\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(- \cos{\left(a \right)} + 1\right)^{2} + \sin^{2}{\left(a \right)}}{2 \sin^{2}{\left(a \right)} + 2 \cos{\left(a \right)} - 2} & \text{otherwise} \end{cases}\right)}{\cos{\left(a \right)}}$$
/ // 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\ |
|1 - |<| 2*cot|-| ||*|<|1 + cot |-| |
| ||< \2/ otherwise || ||< \2/ otherwise |
| |||----------- otherwise || |||------------ otherwise |
| ||| 2/a\ || ||| 2/a\ |
| |||1 + cot |-| || |||-1 + cot |-| |
\ \\\ \2/ // \\\ \2/ /
------------------------------------------------------------------------------------------------------------------
/a pi\
cot|- + --|
\2 4 /
$$\frac{\left(\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + 1\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} + 1}{\cot^{2}{\left(\frac{a}{2} \right)} - 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)}{\cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
// 1 for a mod 2*pi = 0\
|| |
/ // 0 for a mod pi = 0\\ || 2 |
| || || || sin (a) |
| || sin(a) || ||1 + --------- |
| ||----------------------- otherwise || || 4/a\ |
2/a pi\ | ||/ 2 \ || || 4*sin |-| |
2*sin |- + --|*|1 - |<| sin (a) | 2/a\ ||*|< \2/ |
\2 4 / | |||1 + ---------|*sin |-| || ||-------------- otherwise |
| ||| 4/a\| \2/ || || 2 |
| ||| 4*sin |-|| || || sin (a) |
| ||\ \2// || ||-1 + --------- |
\ \\ // || 4/a\ |
|| 4*sin |-| |
\\ \2/ /
-------------------------------------------------------------------------------------------------------
cos(a)
$$\frac{2 \cdot \left(\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{\sin{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \sin^{2}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}}{-1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right) \sin^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\cos{\left(a \right)}}$$
// 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 |- - --| | /a pi\
|1 - |
$$\frac{\left(\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{a}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1}{\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - 1} & \text{otherwise} \end{cases}\right) \cos{\left(\frac{a}{2} - \frac{\pi}{4} \right)}}{\cos{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
// 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 |-| | /a pi\
|1 - |
$$\frac{\left(\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right) \sec{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}}{-1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right) \sec{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\sec{\left(\frac{a}{2} - \frac{\pi}{4} \right)}}$$
// 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 |-- - -| | / a pi\
|1 - |
$$\frac{\left(\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{a}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1}{\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} - 1} & \text{otherwise} \end{cases}\right) \csc{\left(- \frac{a}{2} + \frac{\pi}{4} \right)}}{\csc{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
// / pi\ \
|| zoo for |a + --| mod pi = 0|
// /a pi\ \ || \ 2 / |
|| 0 for |- + --| mod pi = 0| // 1 for a mod 2*pi = 0\ || |
/ // 0 for a mod pi = 0\\ || \2 4 / | || | || /a pi\ |
2*|1 - |< ||*|< |*|< 1 |*|< tan|- + --| |
\ \\sin(a) otherwise // || 2/a pi\ 4/a pi\ | ||------ otherwise | || \2 4 / |
||4*cot |- + --|*sin |- + --| otherwise | \\cos(a) / ||-------------- otherwise |
\\ \4 8 / \4 8 / / || 2/a pi\ |
||2*sin |- + --| |
\\ \2 4 / /
$$2 \cdot \left(\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1\right) \left(\begin{cases} 0 & \text{for}\: \left(\frac{a}{2} + \frac{\pi}{4}\right) \bmod \pi = 0 \\4 \sin^{4}{\left(\frac{a}{4} + \frac{\pi}{8} \right)} \cot^{2}{\left(\frac{a}{4} + \frac{\pi}{8} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\cos{\left(a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{\tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{2 \sin^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}} & \text{otherwise} \end{cases}\right)$$
// /a 7*pi\ \
// /a pi\ \ || 1 for |- + ----| mod 2*pi = 0|
|| 1 for |- + --| mod 2*pi = 0| || \2 4 / |
/ // / 3*pi\ \\ // 1 for a mod 2*pi = 0\ || \2 4 / | || |
| || 1 for |a + ----| mod 2*pi = 0|| || | || | || 2/a 3*pi\ |
|1 - |< \ 2 / ||*|< 1 |*|< 1 |*|<-1 + tan |- + ----| |
| || || ||------ otherwise | ||-------------------------------- otherwise | || \4 8 / |
\ \\sin(a) otherwise // \\cos(a) / ||/ 2/a pi\\ 2/a pi\ | ||------------------- otherwise |
|||-1 + cot |- + --||*sin |- + --| | || 2/a 3*pi\ |
\\\ \4 8 // \4 8 / / || 1 + tan |- + ----| |
\\ \4 8 / /
$$\left(\left(- \begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\cos{\left(a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(\frac{a}{2} + \frac{\pi}{4}\right) \bmod 2 \pi = 0 \\\frac{1}{\left(\cot^{2}{\left(\frac{a}{4} + \frac{\pi}{8} \right)} - 1\right) \sin^{2}{\left(\frac{a}{4} + \frac{\pi}{8} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(\frac{a}{2} + \frac{7 \pi}{4}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{a}{4} + \frac{3 \pi}{8} \right)} - 1}{\tan^{2}{\left(\frac{a}{4} + \frac{3 \pi}{8} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// /a pi\ \
|| 0 for |- + --| mod pi = 0| // / pi\ \
/ // 0 for a mod pi = 0\\ || \2 4 / | // 1 for a mod 2*pi = 0\ || zoo for |a + --| mod pi = 0|
| || || || | || | || \ 2 / |
| || /a\ || || 2/a pi\ | || 2/a\ | || |
| || 2*cot|-| || || 4*cot |- + --| | ||1 + cot |-| | || 2/a pi\ |
2*|1 - |< \2/ ||*|< \4 8 / |*|< \2/ |*|<1 + cot |- + --| |
| ||----------- otherwise || ||------------------- otherwise | ||------------ otherwise | || \2 4 / |
| || 2/a\ || || 2 | || 2/a\ | ||---------------- otherwise |
| ||1 + cot |-| || ||/ 2/a pi\\ | ||-1 + cot |-| | || /a pi\ |
\ \\ \2/ // |||1 + cot |- + --|| | \\ \2/ / || 2*cot|- + --| |
||\ \4 8 // | \\ \2 4 / /
\\ /
$$2 \cdot \left(\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1\right) \left(\begin{cases} 0 & \text{for}\: \left(\frac{a}{2} + \frac{\pi}{4}\right) \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{a}{4} + \frac{\pi}{8} \right)}}{\left(\cot^{2}{\left(\frac{a}{4} + \frac{\pi}{8} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} + 1}{\cot^{2}{\left(\frac{a}{2} \right)} - 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1}{2 \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)}} & \text{otherwise} \end{cases}\right)$$
/ // / 3*pi\ \\ // /a pi\ \ // /a 7*pi\ \ | || 1 for |a + ----| mod 2*pi = 0|| // 1 for a mod 2*pi = 0\ || 1 for |- + --| mod 2*pi = 0| || 1 for |- + ----| mod 2*pi = 0| | || \ 2 / || || | || \2 4 / | || \2 4 / | | || || || 2/a\ | || | || | | || 2/a pi\ || ||1 + cot |-| | || 2/a pi\ | || 2/a 3*pi\ | |1 - |<-1 + tan |- + --| ||*|< \2/ |*|< 1 + cot |- + --| |*|<-1 + tan |- + ----| | | || \2 4 / || ||------------ otherwise | || \4 8 / | || \4 8 / | | ||----------------- otherwise || || 2/a\ | ||----------------- otherwise | ||------------------- otherwise | | || 2/a pi\ || ||-1 + cot |-| | || 2/a pi\ | || 2/a 3*pi\ | | || 1 + tan |- + --| || \\ \2/ / ||-1 + cot |- + --| | || 1 + tan |- + ----| | \ \\ \2 4 / // \\ \4 8 / / \\ \4 8 / /
$$\left(\left(- \begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} + 1}{\cot^{2}{\left(\frac{a}{2} \right)} - 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(\frac{a}{2} + \frac{\pi}{4}\right) \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{4} + \frac{\pi}{8} \right)} + 1}{\cot^{2}{\left(\frac{a}{4} + \frac{\pi}{8} \right)} - 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(\frac{a}{2} + \frac{7 \pi}{4}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{a}{4} + \frac{3 \pi}{8} \right)} - 1}{\tan^{2}{\left(\frac{a}{4} + \frac{3 \pi}{8} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
(1 - Piecewise((1, Mod(a + 3*pi/2 = 2*pi, 0)), ((-1 + tan(a/2 + pi/4)^2)/(1 + tan(a/2 + pi/4)^2), True)))*Piecewise((1, Mod(a = 2*pi, 0)), ((1 + cot(a/2)^2)/(-1 + cot(a/2)^2), True))*Piecewise((1, Mod(a/2 + pi/4 = 2*pi, 0)), ((1 + cot(a/4 + pi/8)^2)/(-1 + cot(a/4 + pi/8)^2), True))*Piecewise((1, Mod(a/2 + 7*pi/4 = 2*pi, 0)), ((-1 + tan(a/4 + 3*pi/8)^2)/(1 + tan(a/4 + 3*pi/8)^2), True))