Господин Экзамен
Общий знаменатель cos(pi/4-a)-cos(pi/4+a)
Решение
Вы ввели
[src]
/pi \ /pi \ cos|-- - a| - cos|-- + a| \4 / \4 /
$$\cos{\left(- a + \frac{\pi}{4} \right)} - \cos{\left(a + \frac{\pi}{4} \right)}$$
cos(pi/4 - a) - cos(pi/4 + a)
Собрать выражение
[src]
/ pi\ / pi\
- cos|a + --| + sin|a + --|
\ 4 / \ 4 /
$$\sin{\left(a + \frac{\pi}{4} \right)} - \cos{\left(a + \frac{\pi}{4} \right)}$$
-cos(a + pi/4) + sin(a + pi/4)
Комбинаторика
[src]
/ pi\ / pi\
- cos|a + --| + sin|a + --|
\ 4 / \ 4 /
$$\sin{\left(a + \frac{\pi}{4} \right)} - \cos{\left(a + \frac{\pi}{4} \right)}$$
-cos(a + pi/4) + sin(a + pi/4)
Степени
[src]
/ pi\ / pi\
- cos|a + --| + sin|a + --|
\ 4 / \ 4 /
$$\sin{\left(a + \frac{\pi}{4} \right)} - \cos{\left(a + \frac{\pi}{4} \right)}$$
/ pi\ / pi\ / pi\ / pi\
I*|a - --| I*|-a + --| I*|a + --| I*|-a - --|
\ 4 / \ 4 / \ 4 / \ 4 /
e e e e
----------- + ------------ - ----------- - ------------
2 2 2 2
$$- \frac{e^{i \left(- a - \frac{\pi}{4}\right)}}{2} + \frac{e^{i \left(- a + \frac{\pi}{4}\right)}}{2} + \frac{e^{i \left(a - \frac{\pi}{4}\right)}}{2} - \frac{e^{i \left(a + \frac{\pi}{4}\right)}}{2}$$
exp(i*(a - pi/4))/2 + exp(i*(-a + pi/4))/2 - exp(i*(a + pi/4))/2 - exp(i*(-a - pi/4))/2
Объединение рациональных выражений
[src]
/pi + 4*a\ /pi - 4*a\
- cos|--------| + cos|--------|
\ 4 / \ 4 /
$$\cos{\left(\frac{- 4 a + \pi}{4} \right)} - \cos{\left(\frac{4 a + \pi}{4} \right)}$$
-cos((pi + 4*a)/4) + cos((pi - 4*a)/4)
Общий знаменатель
[src]
/ pi\ / pi\
- cos|a + --| + sin|a + --|
\ 4 / \ 4 /
$$\sin{\left(a + \frac{\pi}{4} \right)} - \cos{\left(a + \frac{\pi}{4} \right)}$$
-cos(a + pi/4) + sin(a + pi/4)
Рациональный знаменатель
[src]
/ pi\ / pi\
- cos|a + --| + sin|a + --|
\ 4 / \ 4 /
$$\sin{\left(a + \frac{\pi}{4} \right)} - \cos{\left(a + \frac{\pi}{4} \right)}$$
-cos(a + pi/4) + sin(a + pi/4)
Тригонометрическая часть
[src]
___ \/ 2 *sin(a)
$$\sqrt{2} \sin{\left(a \right)}$$
___ \/ 2 ------ csc(a)
$$\frac{\sqrt{2}}{\csc{\left(a \right)}}$$
___ / pi\
\/ 2 *cos|a - --|
\ 2 /
$$\sqrt{2} \cos{\left(a - \frac{\pi}{2} \right)}$$
___ \/ 2 ----------- / pi\ sec|a - --| \ 2 /
$$\frac{\sqrt{2}}{\sec{\left(a - \frac{\pi}{2} \right)}}$$
/ pi\ / pi\
- cos|a + --| + sin|a + --|
\ 4 / \ 4 /
$$\sin{\left(a + \frac{\pi}{4} \right)} - \cos{\left(a + \frac{\pi}{4} \right)}$$
/ pi\ / pi\
- cos|a + --| + cos|a - --|
\ 4 / \ 4 /
$$\cos{\left(a - \frac{\pi}{4} \right)} - \cos{\left(a + \frac{\pi}{4} \right)}$$
/ ___\
2*\1 - \/ 2 /*sin(a)
--------------------
___
-2 + \/ 2
$$\frac{2 \cdot \left(- \sqrt{2} + 1\right) \sin{\left(a \right)}}{-2 + \sqrt{2}}$$
/ 3*pi\ / pi\
- sin|a + ----| + sin|a + --|
\ 4 / \ 4 /
$$\sin{\left(a + \frac{\pi}{4} \right)} - \sin{\left(a + \frac{3 \pi}{4} \right)}$$
___ /a\
2*\/ 2 *tan|-|
\2/
--------------
2/a\
1 + tan |-|
\2/
$$\frac{2 \sqrt{2} \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}$$
1 1 ----------- - ------------ / pi\ / pi\ csc|a + --| csc|-a + --| \ 4 / \ 4 /
$$\frac{1}{\csc{\left(a + \frac{\pi}{4} \right)}} - \frac{1}{\csc{\left(- a + \frac{\pi}{4} \right)}}$$
1 1 ----------- - ----------- / pi\ / pi\ csc|a + --| sec|a + --| \ 4 / \ 4 /
$$- \frac{1}{\sec{\left(a + \frac{\pi}{4} \right)}} + \frac{1}{\csc{\left(a + \frac{\pi}{4} \right)}}$$
1 1 ----------- - ----------- / pi\ / pi\ sec|a - --| sec|a + --| \ 4 / \ 4 /
$$- \frac{1}{\sec{\left(a + \frac{\pi}{4} \right)}} + \frac{1}{\sec{\left(a - \frac{\pi}{4} \right)}}$$
1 1 ------------ - ----------- / pi\ / pi\ sec|-a + --| sec|a + --| \ 4 / \ 4 /
$$- \frac{1}{\sec{\left(a + \frac{\pi}{4} \right)}} + \frac{1}{\sec{\left(- a + \frac{\pi}{4} \right)}}$$
1 1 -------------- - ------------ / 3*pi\ / pi\ csc|-a + ----| csc|-a + --| \ 4 / \ 4 /
$$\frac{1}{\csc{\left(- a + \frac{3 \pi}{4} \right)}} - \frac{1}{\csc{\left(- a + \frac{\pi}{4} \right)}}$$
___ // 0 for a mod pi = 0\
\/ 2 *|< |
\\sin(a) otherwise /
$$\sqrt{2} \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right)$$
2/a\ / ___\ /a\
4*cos |-|*\1 - \/ 2 /*tan|-|
\2/ \2/
----------------------------
___
-2 + \/ 2
$$\frac{4 \cdot \left(- \sqrt{2} + 1\right) \cos^{2}{\left(\frac{a}{2} \right)} \tan{\left(\frac{a}{2} \right)}}{-2 + \sqrt{2}}$$
/ ___\ /a\ /a pi\
4*\1 - \/ 2 /*cos|-|*cos|- - --|
\2/ \2 2 /
--------------------------------
___
-2 + \/ 2
$$\frac{4 \cdot \left(- \sqrt{2} + 1\right) \cos{\left(\frac{a}{2} \right)} \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{-2 + \sqrt{2}}$$
/ ___\
4*\1 - \/ 2 /
-------------------------------
/ ___\ /a\ /a pi\
\-2 + \/ 2 /*sec|-|*sec|- - --|
\2/ \2 2 /
$$\frac{4 \cdot \left(- \sqrt{2} + 1\right)}{\left(-2 + \sqrt{2}\right) \sec{\left(\frac{a}{2} \right)} \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}$$
/ ___\
4*\1 - \/ 2 /
-------------------------------
/ ___\ /a\ /pi a\
\-2 + \/ 2 /*csc|-|*csc|-- - -|
\2/ \2 2/
$$\frac{4 \cdot \left(- \sqrt{2} + 1\right)}{\left(-2 + \sqrt{2}\right) \csc{\left(\frac{a}{2} \right)} \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}$$
2/a\ 2/pi a\ / ___\
8*sin |-|*sin |-- + -|*\1 - \/ 2 /
\2/ \2 2/
----------------------------------
/ ___\
\-2 + \/ 2 /*sin(a)
$$\frac{8 \cdot \left(- \sqrt{2} + 1\right) \sin^{2}{\left(\frac{a}{2} \right)} \sin^{2}{\left(\frac{a}{2} + \frac{\pi}{2} \right)}}{\left(-2 + \sqrt{2}\right) \sin{\left(a \right)}}$$
// 0 for a mod pi = 0\
|| |
|| /a\ |
___ || 2*cot|-| |
\/ 2 *|< \2/ |
||----------- otherwise |
|| 2/a\ |
||1 + cot |-| |
\\ \2/ /
$$\sqrt{2} \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
2
/ 2/a\\ 4/a\ / ___\ /a\
4*|1 - tan |-|| *cos |-|*\1 - \/ 2 /*tan|-|
\ \4// \4/ \2/
-------------------------------------------
___
-2 + \/ 2
$$\frac{4 \cdot \left(- \sqrt{2} + 1\right) \left(- \tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2} \cos^{4}{\left(\frac{a}{4} \right)} \tan{\left(\frac{a}{2} \right)}}{-2 + \sqrt{2}}$$
2
/ 2/a\\ / ___\ /a\
4*|1 - tan |-|| *\1 - \/ 2 /*tan|-|
\ \4// \2/
-----------------------------------
2
/ 2/a\\ / ___\
|1 + tan |-|| *\-2 + \/ 2 /
\ \4//
$$\frac{4 \cdot \left(- \sqrt{2} + 1\right) \left(- \tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2} \tan{\left(\frac{a}{2} \right)}}{\left(-2 + \sqrt{2}\right) \left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2}}$$
/a 3*pi\
2*tan|- + ----|
\2 8 / 2/a pi\ /a pi\
- ------------------ + 2*sin |- + --|*cot|- + --|
2/a 3*pi\ \2 8 / \2 8 /
1 + tan |- + ----|
\2 8 /
$$2 \sin^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} \cot{\left(\frac{a}{2} + \frac{\pi}{8} \right)} - \frac{2 \tan{\left(\frac{a}{2} + \frac{3 \pi}{8} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{3 \pi}{8} \right)} + 1}$$
// a \
|| 1 for - mod 2*pi = 0|
/ ___\ || 2 | /a\
4*\1 - \/ 2 /*|< |*tan|-|
||1 + cos(a) | \2/
||---------- otherwise |
\\ 2 /
------------------------------------------------------
___
-2 + \/ 2
$$\frac{4 \cdot \left(- \sqrt{2} + 1\right) \left(\begin{cases} 1 & \text{for}\: \frac{a}{2} \bmod 2 \pi = 0 \\\frac{\cos{\left(a \right)} + 1}{2} & \text{otherwise} \end{cases}\right) \tan{\left(\frac{a}{2} \right)}}{-2 + \sqrt{2}}$$
/a 3*pi\ /a pi\
2*tan|- + ----| 2*cot|- + --|
\2 8 / \2 8 /
- ------------------ + ----------------
2/a 3*pi\ 2/a pi\
1 + tan |- + ----| 1 + cot |- + --|
\2 8 / \2 8 /
$$\frac{2 \cot{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} + 1} - \frac{2 \tan{\left(\frac{a}{2} + \frac{3 \pi}{8} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{3 \pi}{8} \right)} + 1}$$
/a 3*pi\ /a pi\
2*tan|- + ----| 2*tan|- + --|
\2 8 / \2 8 /
- ------------------ + ----------------
2/a 3*pi\ 2/a pi\
1 + tan |- + ----| 1 + tan |- + --|
\2 8 / \2 8 /
$$- \frac{2 \tan{\left(\frac{a}{2} + \frac{3 \pi}{8} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{3 \pi}{8} \right)} + 1} + \frac{2 \tan{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} + 1}$$
2/a pi\ /a pi\
1 - tan |- + --| 2*tan|- + --|
\2 8 / \2 8 /
- ---------------- + ----------------
2/a pi\ 2/a pi\
1 + tan |- + --| 1 + tan |- + --|
\2 8 / \2 8 /
$$- \frac{- \tan^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} + 1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} + 1} + \frac{2 \tan{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} + 1}$$
2/a 3*pi\ 2/a pi\
-1 + tan |- + ----| -1 + cot |- + --|
\2 8 / \2 8 /
------------------- - -----------------
2/a 3*pi\ 2/a pi\
1 + tan |- + ----| 1 + cot |- + --|
\2 8 / \2 8 /
$$\frac{\tan^{2}{\left(\frac{a}{2} + \frac{3 \pi}{8} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} + \frac{3 \pi}{8} \right)} + 1} - \frac{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} + 1}$$
2/a 3*pi\
-1 + tan |- + ----|
\2 8 / 2/a pi\ / 2/a pi\\
------------------- - sin |- + --|*|-1 + cot |- + --||
2/a 3*pi\ \2 8 / \ \2 8 //
1 + tan |- + ----|
\2 8 /
$$- \left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} - 1\right) \sin^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} + \frac{\tan^{2}{\left(\frac{a}{2} + \frac{3 \pi}{8} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} + \frac{3 \pi}{8} \right)} + 1}$$
2/a pi\
1 - tan |- + --|
2/a 3*pi\ / 2/a 3*pi\\ \2 8 /
sin |- + ----|*|1 - cot |- + ----|| - ----------------
\2 8 / \ \2 8 // 2/a pi\
1 + tan |- + --|
\2 8 /
$$\left(- \cot^{2}{\left(\frac{a}{2} + \frac{3 \pi}{8} \right)} + 1\right) \sin^{2}{\left(\frac{a}{2} + \frac{3 \pi}{8} \right)} - \frac{- \tan^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} + 1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} + 1}$$
2/a 3*pi\ 2/a pi\
1 - cot |- + ----| 1 - tan |- + --|
\2 8 / \2 8 /
------------------ - ----------------
2/a 3*pi\ 2/a pi\
1 + cot |- + ----| 1 + tan |- + --|
\2 8 / \2 8 /
$$- \frac{- \tan^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} + 1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} + 1} + \frac{- \cot^{2}{\left(\frac{a}{2} + \frac{3 \pi}{8} \right)} + 1}{\cot^{2}{\left(\frac{a}{2} + \frac{3 \pi}{8} \right)} + 1}$$
1
1 - ------------
2/a pi\
cot |- + --|
\2 8 / 2
- ---------------- + ------------------------------
1 / 1 \ /a pi\
1 + ------------ |1 + ------------|*cot|- + --|
2/a pi\ | 2/a pi\| \2 8 /
cot |- + --| | cot |- + --||
\2 8 / \ \2 8 //
$$- \frac{1 - \frac{1}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}}{1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}} + \frac{2}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}\right) \cot{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}$$
2/a pi\ /a 3*pi\
1 - tan |- + --| 2*cos|- - ----|
\2 8 / \2 8 /
- ---------------- + ------------------------------
2/a pi\ / 2/a pi\\ /a pi\
1 + tan |- + --| |1 + tan |- + --||*cos|- + --|
\2 8 / \ \2 8 // \2 8 /
$$- \frac{- \tan^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} + 1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} + 1} + \frac{2 \cos{\left(\frac{a}{2} - \frac{3 \pi}{8} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} + 1\right) \cos{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}$$
2/a pi\ /a pi\
1 - tan |- + --| 2*sec|- + --|
\2 8 / \2 8 /
- ---------------- + --------------------------------
2/a pi\ / 2/a pi\\ /a 3*pi\
1 + tan |- + --| |1 + tan |- + --||*sec|- - ----|
\2 8 / \ \2 8 // \2 8 /
$$- \frac{- \tan^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} + 1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} + 1} + \frac{2 \sec{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} + 1\right) \sec{\left(\frac{a}{2} - \frac{3 \pi}{8} \right)}}$$
2/a pi\ / a 3*pi\
1 - tan |- + --| 2*csc|- - + ----|
\2 8 / \ 2 8 /
- ---------------- + ------------------------------
2/a pi\ / 2/a pi\\ /a pi\
1 + tan |- + --| |1 + tan |- + --||*csc|- + --|
\2 8 / \ \2 8 // \2 8 /
$$- \frac{- \tan^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} + 1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} + 1} + \frac{2 \csc{\left(- \frac{a}{2} + \frac{3 \pi}{8} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} + 1\right) \csc{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}$$
// a \
|| 1 for - mod 2*pi = 0|
|| 2 |
|| |
|| 2 |
/ ___\ ||/ 2/a\\ |
4*\1 - \/ 2 /*|<|-1 + cot |-|| |
||\ \4// |
||--------------- otherwise |
|| 2 |
|| / 2/a\\ |
|| |1 + cot |-|| |
\\ \ \4// /
----------------------------------------------------
/ ___\ /a\
\-2 + \/ 2 /*cot|-|
\2/
$$\frac{4 \cdot \left(- \sqrt{2} + 1\right) \left(\begin{cases} 1 & \text{for}\: \frac{a}{2} \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)}{\left(-2 + \sqrt{2}\right) \cot{\left(\frac{a}{2} \right)}}$$
4/a pi\
8*sin |- + --|
\2 8 / 2/a pi\
1 - -------------- 4*sin |- + --|
1 + sin(2*a) \2 8 /
- ------------------ + --------------------------------
4/a pi\ / 4/a pi\\
8*sin |- + --| | 8*sin |- + --||
\2 8 / | \2 8 /| / pi\
1 + -------------- |1 + --------------|*sin|a + --|
1 + sin(2*a) \ 1 + sin(2*a) / \ 4 /
$$- \frac{- \frac{8 \sin^{4}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}{\sin{\left(2 a \right)} + 1} + 1}{\frac{8 \sin^{4}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}{\sin{\left(2 a \right)} + 1} + 1} + \frac{4 \sin^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}{\left(\frac{8 \sin^{4}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}{\sin{\left(2 a \right)} + 1} + 1\right) \sin{\left(a + \frac{\pi}{4} \right)}}$$
4/a pi\
4*sin |- + --|
\2 8 /
1 - --------------
2/ pi\ 2/a pi\
sin |a + --| 4*sin |- + --|
\ 4 / \2 8 /
- ------------------ + --------------------------------
4/a pi\ / 4/a pi\\
4*sin |- + --| | 4*sin |- + --||
\2 8 / | \2 8 /| / pi\
1 + -------------- |1 + --------------|*sin|a + --|
2/ pi\ | 2/ pi\ | \ 4 /
sin |a + --| | sin |a + --| |
\ 4 / \ \ 4 / /
$$- \frac{- \frac{4 \sin^{4}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}{\sin^{2}{\left(a + \frac{\pi}{4} \right)}} + 1}{\frac{4 \sin^{4}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}{\sin^{2}{\left(a + \frac{\pi}{4} \right)}} + 1} + \frac{4 \sin^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}{\sin^{2}{\left(a + \frac{\pi}{4} \right)}} + 1\right) \sin{\left(a + \frac{\pi}{4} \right)}}$$
// / 3*pi\ \ // / pi\ \ || 0 for |a + ----| mod pi = 0| || 0 for |a + --| mod pi = 0| || \ 4 / | || \ 4 / | - |< | + |< | || 2/a 3*pi\ /a 3*pi\ | || 2/a pi\ /a pi\ | ||2*sin |- + ----|*cot|- + ----| otherwise | ||2*sin |- + --|*cot|- + --| otherwise | \\ \2 8 / \2 8 / / \\ \2 8 / \2 8 / /
$$\left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod \pi = 0 \\2 \sin^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} \cot{\left(\frac{a}{2} + \frac{\pi}{8} \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 0 & \text{for}\: \left(a + \frac{3 \pi}{4}\right) \bmod \pi = 0 \\2 \sin^{2}{\left(\frac{a}{2} + \frac{3 \pi}{8} \right)} \cot{\left(\frac{a}{2} + \frac{3 \pi}{8} \right)} & \text{otherwise} \end{cases}\right)$$
// / pi\ \ || 1 for |a + --| mod 2*pi = 0| || \ 4 / | // / pi\ \ || | || 0 for |a + --| mod pi = 0| || / 2/ pi\ \ | || \ 4 / | - |< | sin |a + --| | | + |< | || 2/a pi\ | \ 4 / | | || / pi\ | ||sin |- + --|*|-1 + --------------| otherwise | ||sin|a + --| otherwise | || \2 8 / | 4/a pi\| | \\ \ 4 / / || | 4*sin |- + --|| | \\ \ \2 8 // /
$$\left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod \pi = 0 \\\sin{\left(a + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod 2 \pi = 0 \\\left(-1 + \frac{\sin^{2}{\left(a + \frac{\pi}{4} \right)}}{4 \sin^{4}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}\right) \sin^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} & \text{otherwise} \end{cases}\right)$$
// / pi\ \ // / pi\ \ || 1 for |a + --| mod 2*pi = 0| || 0 for |a + --| mod pi = 0| || \ 4 / | || \ 4 / | - |< | + |< | || 2/a pi\ / 2/a pi\\ | || 2/a pi\ /a pi\ | ||sin |- + --|*|-1 + cot |- + --|| otherwise | ||2*sin |- + --|*cot|- + --| otherwise | \\ \2 8 / \ \2 8 // / \\ \2 8 / \2 8 / /
$$\left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod \pi = 0 \\2 \sin^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} \cot{\left(\frac{a}{2} + \frac{\pi}{8} \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} - 1\right) \sin^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} & \text{otherwise} \end{cases}\right)$$
// / 3*pi\ \ // / pi\ \ || 0 for |a + ----| mod pi = 0| || 0 for |a + --| mod pi = 0| || \ 4 / | || \ 4 / | || | || | || /a 3*pi\ | || /a pi\ | - |< 2*cot|- + ----| | + |< 2*cot|- + --| | || \2 8 / | || \2 8 / | ||------------------ otherwise | ||---------------- otherwise | || 2/a 3*pi\ | || 2/a pi\ | ||1 + cot |- + ----| | ||1 + cot |- + --| | \\ \2 8 / / \\ \2 8 / /
$$\left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} + 1} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 0 & \text{for}\: \left(a + \frac{3 \pi}{4}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} + \frac{3 \pi}{8} \right)}}{\cot^{2}{\left(\frac{a}{2} + \frac{3 \pi}{8} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// / pi\ \ // / pi\ \ || 1 for |a + --| mod 2*pi = 0| || 0 for |a + --| mod pi = 0| || \ 4 / | || \ 4 / | || | || | || 2/a pi\ | || /a pi\ | - |<-1 + cot |- + --| | + |< 2*cot|- + --| | || \2 8 / | || \2 8 / | ||----------------- otherwise | ||---------------- otherwise | || 2/a pi\ | || 2/a pi\ | || 1 + cot |- + --| | ||1 + cot |- + --| | \\ \2 8 / / \\ \2 8 / /
$$\left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} + 1} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// / pi\ \ || 1 for |a + --| mod 2*pi = 0| // / pi\ \ || \ 4 / | || 0 for |a + --| mod pi = 0| || | || \ 4 / | - |< 2/a 3*pi\ / 1 \ | + |< | ||cos |- - ----|*|-1 + ------------| otherwise | || /a 3*pi\ /a pi\ | || \2 8 / | 2/a pi\| | ||2*cos|- - ----|*cos|- + --| otherwise | || | tan |- + --|| | \\ \2 8 / \2 8 / / \\ \ \2 8 // /
$$\left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod \pi = 0 \\2 \cos{\left(\frac{a}{2} - \frac{3 \pi}{8} \right)} \cos{\left(\frac{a}{2} + \frac{\pi}{8} \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod 2 \pi = 0 \\\left(-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}\right) \cos^{2}{\left(\frac{a}{2} - \frac{3 \pi}{8} \right)} & \text{otherwise} \end{cases}\right)$$
// / pi\ \ || 1 for |a + --| mod 2*pi = 0| || \ 4 / | // / pi\ \ || | || 0 for |a + --| mod pi = 0| || 1 | || \ 4 / | ||-1 + ------------ | || | - |< 2/a pi\ | + |< 2 | || tan |- + --| | ||------------------------- otherwise | || \2 8 / | || /a 3*pi\ /a pi\ | ||----------------- otherwise | ||sec|- - ----|*sec|- + --| | || 2/a 3*pi\ | \\ \2 8 / \2 8 / / || sec |- - ----| | \\ \2 8 / /
$$\left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod \pi = 0 \\\frac{2}{\sec{\left(\frac{a}{2} - \frac{3 \pi}{8} \right)} \sec{\left(\frac{a}{2} + \frac{\pi}{8} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}}{\sec^{2}{\left(\frac{a}{2} - \frac{3 \pi}{8} \right)}} & \text{otherwise} \end{cases}\right)$$
// / pi\ \ || 1 for |a + --| mod 2*pi = 0| || \ 4 / | // / pi\ \ || | || 0 for |a + --| mod pi = 0| || 1 | || \ 4 / | ||-1 + ------------ | || | - |< 2/a pi\ | + |< 2 | || tan |- + --| | ||--------------------------- otherwise | || \2 8 / | || /a pi\ / a 3*pi\ | ||----------------- otherwise | ||csc|- + --|*csc|- - + ----| | || 2/a pi\ | \\ \2 8 / \ 2 8 / / || csc |- + --| | \\ \2 8 / /
$$\left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod \pi = 0 \\\frac{2}{\csc{\left(- \frac{a}{2} + \frac{3 \pi}{8} \right)} \csc{\left(\frac{a}{2} + \frac{\pi}{8} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}}{\csc^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}} & \text{otherwise} \end{cases}\right)$$
2/a pi\
sec |- + --|
\2 8 /
1 - --------------
2/a 3*pi\ /a pi\
sec |- - ----| 2*sec|- + --|
\2 8 / \2 8 /
- ------------------ + ----------------------------------
2/a pi\ / 2/a pi\ \
sec |- + --| | sec |- + --| |
\2 8 / | \2 8 / | /a 3*pi\
1 + -------------- |1 + --------------|*sec|- - ----|
2/a 3*pi\ | 2/a 3*pi\| \2 8 /
sec |- - ----| | sec |- - ----||
\2 8 / \ \2 8 //
$$- \frac{1 - \frac{\sec^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{3 \pi}{8} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{3 \pi}{8} \right)}}} + \frac{2 \sec{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{3 \pi}{8} \right)}}\right) \sec{\left(\frac{a}{2} - \frac{3 \pi}{8} \right)}}$$
2/a 3*pi\
cos |- - ----|
\2 8 /
1 - --------------
2/a pi\ /a 3*pi\
cos |- + --| 2*cos|- - ----|
\2 8 / \2 8 /
- ------------------ + --------------------------------
2/a 3*pi\ / 2/a 3*pi\\
cos |- - ----| | cos |- - ----||
\2 8 / | \2 8 /| /a pi\
1 + -------------- |1 + --------------|*cos|- + --|
2/a pi\ | 2/a pi\ | \2 8 /
cos |- + --| | cos |- + --| |
\2 8 / \ \2 8 / /
$$- \frac{- \frac{\cos^{2}{\left(\frac{a}{2} - \frac{3 \pi}{8} \right)}}{\cos^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}} + 1}{\frac{\cos^{2}{\left(\frac{a}{2} - \frac{3 \pi}{8} \right)}}{\cos^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}} + 1} + \frac{2 \cos{\left(\frac{a}{2} - \frac{3 \pi}{8} \right)}}{\left(\frac{\cos^{2}{\left(\frac{a}{2} - \frac{3 \pi}{8} \right)}}{\cos^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}} + 1\right) \cos{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}$$
2/ a 3*pi\
csc |- - + ----|
\ 2 8 /
1 - ----------------
2/a pi\ / a 3*pi\
csc |- + --| 2*csc|- - + ----|
\2 8 / \ 2 8 /
- -------------------- + ----------------------------------
2/ a 3*pi\ / 2/ a 3*pi\\
csc |- - + ----| | csc |- - + ----||
\ 2 8 / | \ 2 8 /| /a pi\
1 + ---------------- |1 + ----------------|*csc|- + --|
2/a pi\ | 2/a pi\ | \2 8 /
csc |- + --| | csc |- + --| |
\2 8 / \ \2 8 / /
$$- \frac{- \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{3 \pi}{8} \right)}}{\csc^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}} + 1}{\frac{\csc^{2}{\left(- \frac{a}{2} + \frac{3 \pi}{8} \right)}}{\csc^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}} + 1} + \frac{2 \csc{\left(- \frac{a}{2} + \frac{3 \pi}{8} \right)}}{\left(\frac{\csc^{2}{\left(- \frac{a}{2} + \frac{3 \pi}{8} \right)}}{\csc^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}} + 1\right) \csc{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}$$
// / pi\ \ // / 7*pi\ \ || 1 for |a + --| mod 2*pi = 0| || 1 for |a + ----| mod 2*pi = 0| || \ 4 / | || \ 4 / | || | || | || 2/a pi\ | || 2/a 3*pi\ | - |<-1 + cot |- + --| | + |<-1 + tan |- + ----| | || \2 8 / | || \2 8 / | ||----------------- otherwise | ||------------------- otherwise | || 2/a pi\ | || 2/a 3*pi\ | || 1 + cot |- + --| | || 1 + tan |- + ----| | \\ \2 8 / / \\ \2 8 / /
$$\left(- \begin{cases} 1 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{7 \pi}{4}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{a}{2} + \frac{3 \pi}{8} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} + \frac{3 \pi}{8} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// / 7*pi\ \
|| 1 for |a + ----| mod 2*pi = 0|
// / pi\ \ || \ 4 / |
|| 1 for |a + --| mod 2*pi = 0| || |
|| \ 4 / | || 2/a 3*pi\ |
- |< | + |<-1 + tan |- + ----| |
|| 2/a pi\ / 2/a pi\\ | || \2 8 / |
||sin |- + --|*|-1 + cot |- + --|| otherwise | ||------------------- otherwise |
\\ \2 8 / \ \2 8 // / || 2/a 3*pi\ |
|| 1 + tan |- + ----| |
\\ \2 8 / /
$$\left(- \begin{cases} 1 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} - 1\right) \sin^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{7 \pi}{4}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{a}{2} + \frac{3 \pi}{8} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} + \frac{3 \pi}{8} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// / pi\ \ || 1 for |a + --| mod 2*pi = 0| || \ 4 / | // / pi\ \ || | || 0 for |a + --| mod pi = 0| || 1 | || \ 4 / | ||-1 + ------------ | || | || 2/a pi\ | || 2 | - |< tan |- + --| | + |<------------------------------ otherwise | || \2 8 / | ||/ 1 \ /a pi\ | ||----------------- otherwise | |||1 + ------------|*tan|- + --| | || 1 | ||| 2/a pi\| \2 8 / | || 1 + ------------ | ||| tan |- + --|| | || 2/a pi\ | \\\ \2 8 // / || tan |- + --| | \\ \2 8 / /
$$\left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}\right) \tan{\left(\frac{a}{2} + \frac{\pi}{8} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}} & \text{otherwise} \end{cases}\right)$$
// / pi\ \ || 1 for |a + --| mod 2*pi = 0| || \ 4 / | // / pi\ \ || | || 0 for |a + --| mod pi = 0| || / 2/a pi\ \ | || \ 4 / | - |< | cos |- + --| | | + |< | || 2/a 3*pi\ | \2 8 / | | || /a 3*pi\ /a pi\ | ||cos |- - ----|*|-1 + --------------| otherwise | ||2*cos|- - ----|*cos|- + --| otherwise | || \2 8 / | 2/a 3*pi\| | \\ \2 8 / \2 8 / / || | cos |- - ----|| | \\ \ \2 8 // /
$$\left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod \pi = 0 \\2 \cos{\left(\frac{a}{2} - \frac{3 \pi}{8} \right)} \cos{\left(\frac{a}{2} + \frac{\pi}{8} \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod 2 \pi = 0 \\\left(-1 + \frac{\cos^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{3 \pi}{8} \right)}}\right) \cos^{2}{\left(\frac{a}{2} - \frac{3 \pi}{8} \right)} & \text{otherwise} \end{cases}\right)$$
// / pi\ \ || 1 for |a + --| mod 2*pi = 0| || \ 4 / | || | // / pi\ \ || 2/a 3*pi\ | || 0 for |a + --| mod pi = 0| || sec |- - ----| | || \ 4 / | || \2 8 / | || | - |<-1 + -------------- | + |< 2 | || 2/a pi\ | ||------------------------- otherwise | || sec |- + --| | || /a 3*pi\ /a pi\ | || \2 8 / | ||sec|- - ----|*sec|- + --| | ||------------------- otherwise | \\ \2 8 / \2 8 / / || 2/a 3*pi\ | || sec |- - ----| | \\ \2 8 / /
$$\left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod \pi = 0 \\\frac{2}{\sec{\left(\frac{a}{2} - \frac{3 \pi}{8} \right)} \sec{\left(\frac{a}{2} + \frac{\pi}{8} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod 2 \pi = 0 \\\frac{\frac{\sec^{2}{\left(\frac{a}{2} - \frac{3 \pi}{8} \right)}}{\sec^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}} - 1}{\sec^{2}{\left(\frac{a}{2} - \frac{3 \pi}{8} \right)}} & \text{otherwise} \end{cases}\right)$$
// / pi\ \ || 1 for |a + --| mod 2*pi = 0| || \ 4 / | || | // / pi\ \ || 2/a pi\ | || 0 for |a + --| mod pi = 0| || csc |- + --| | || \ 4 / | || \2 8 / | || | - |<-1 + ---------------- | + |< 2 | || 2/ a 3*pi\ | ||--------------------------- otherwise | || csc |- - + ----| | || /a pi\ / a 3*pi\ | || \ 2 8 / | ||csc|- + --|*csc|- - + ----| | ||--------------------- otherwise | \\ \2 8 / \ 2 8 / / || 2/a pi\ | || csc |- + --| | \\ \2 8 / /
$$\left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod \pi = 0 \\\frac{2}{\csc{\left(- \frac{a}{2} + \frac{3 \pi}{8} \right)} \csc{\left(\frac{a}{2} + \frac{\pi}{8} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod 2 \pi = 0 \\\frac{-1 + \frac{\csc^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{3 \pi}{8} \right)}}}{\csc^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}} & \text{otherwise} \end{cases}\right)$$
// / pi\ \ || 1 for |a + --| mod 2*pi = 0| // / pi\ \ || \ 4 / | || 0 for |a + --| mod pi = 0| || | || \ 4 / | || 1 | || | ||-1 + ------------ | || /a 3*pi\ | || 2/a pi\ | || 2*sec|- - ----| | - |< tan |- + --| | + |< \2 8 / | || \2 8 / | ||------------------------------ otherwise | ||----------------- otherwise | ||/ 1 \ /a pi\ | || 1 | |||1 + ------------|*sec|- + --| | || 1 + ------------ | ||| 2/a pi\| \2 8 / | || 2/a pi\ | ||| tan |- + --|| | || tan |- + --| | \\\ \2 8 // / \\ \2 8 / /
$$\left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{a}{2} - \frac{3 \pi}{8} \right)}}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}\right) \sec{\left(\frac{a}{2} + \frac{\pi}{8} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}} & \text{otherwise} \end{cases}\right)$$
// / pi\ \ || 1 for |a + --| mod 2*pi = 0| // / pi\ \ || \ 4 / | || 0 for |a + --| mod pi = 0| || | || \ 4 / | || 1 | || | ||-1 + ------------ | || /a pi\ | || 2/a pi\ | || 2*cos|- + --| | - |< tan |- + --| | + |< \2 8 / | || \2 8 / | ||-------------------------------- otherwise | ||----------------- otherwise | ||/ 1 \ /a 3*pi\ | || 1 | |||1 + ------------|*cos|- - ----| | || 1 + ------------ | ||| 2/a pi\| \2 8 / | || 2/a pi\ | ||| tan |- + --|| | || tan |- + --| | \\\ \2 8 // / \\ \2 8 / /
$$\left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}\right) \cos{\left(\frac{a}{2} - \frac{3 \pi}{8} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}} & \text{otherwise} \end{cases}\right)$$
// / pi\ \ || 1 for |a + --| mod 2*pi = 0| // / pi\ \ || \ 4 / | || 0 for |a + --| mod pi = 0| || | || \ 4 / | || 1 | || | ||-1 + ------------ | || /a pi\ | || 2/a pi\ | || 2*csc|- + --| | - |< tan |- + --| | + |< \2 8 / | || \2 8 / | ||---------------------------------- otherwise | ||----------------- otherwise | ||/ 1 \ / a 3*pi\ | || 1 | |||1 + ------------|*csc|- - + ----| | || 1 + ------------ | ||| 2/a pi\| \ 2 8 / | || 2/a pi\ | ||| tan |- + --|| | || tan |- + --| | \\\ \2 8 // / \\ \2 8 / /
$$\left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}\right) \csc{\left(- \frac{a}{2} + \frac{3 \pi}{8} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}} & \text{otherwise} \end{cases}\right)$$
// / pi\ \ || 1 for |a + --| mod 2*pi = 0| // / pi\ \ || \ 4 / | || 0 for |a + --| mod pi = 0| || | || \ 4 / | || 2/a pi\ / 1 \ | || | ||4*tan |- + --|*|-1 + ------------| | || 2/a pi\ | || \4 16/ | 2/a pi\| | || 8*tan |- + --| | - |< | tan |- + --|| | + |< \4 16/ | || \ \2 8 // | ||------------------------------- otherwise | ||---------------------------------- otherwise | || 2 | || 2 | ||/ 2/a pi\\ /a pi\ | || / 2/a pi\\ | |||1 + tan |- + --|| *tan|- + --| | || |1 + tan |- + --|| | ||\ \4 16// \2 8 / | || \ \4 16// | \\ / \\ /
$$\left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod \pi = 0 \\\frac{8 \tan^{2}{\left(\frac{a}{4} + \frac{\pi}{16} \right)}}{\left(\tan^{2}{\left(\frac{a}{4} + \frac{\pi}{16} \right)} + 1\right)^{2} \tan{\left(\frac{a}{2} + \frac{\pi}{8} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod 2 \pi = 0 \\\frac{4 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}\right) \tan^{2}{\left(\frac{a}{4} + \frac{\pi}{16} \right)}}{\left(\tan^{2}{\left(\frac{a}{4} + \frac{\pi}{16} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// / pi\ \ || 1 for |a + --| mod 2*pi = 0| || \ 4 / | // / pi\ \ || | || 0 for |a + --| mod pi = 0| || 2/ pi\ | || \ 4 / | || sin |a + --| | || | || \ 4 / | || / pi\ | ||-1 + -------------- | || sin|a + --| | || 4/a pi\ | || \ 4 / | - |< 4*sin |- + --| | + |<--------------------------------- otherwise | || \2 8 / | ||/ 2/ pi\ \ | ||------------------- otherwise | ||| sin |a + --| | | || 2/ pi\ | ||| \ 4 / | 2/a pi\ | || sin |a + --| | |||1 + --------------|*sin |- + --| | || \ 4 / | ||| 4/a pi\| \2 8 / | || 1 + -------------- | ||| 4*sin |- + --|| | || 4/a pi\ | \\\ \2 8 // / || 4*sin |- + --| | \\ \2 8 / /
$$\left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod \pi = 0 \\\frac{\sin{\left(a + \frac{\pi}{4} \right)}}{\left(1 + \frac{\sin^{2}{\left(a + \frac{\pi}{4} \right)}}{4 \sin^{4}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}\right) \sin^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(a + \frac{\pi}{4} \right)}}{4 \sin^{4}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}}{1 + \frac{\sin^{2}{\left(a + \frac{\pi}{4} \right)}}{4 \sin^{4}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}} & \text{otherwise} \end{cases}\right)$$
// / pi\ \ || 1 for |a + --| mod 2*pi = 0| || \ 4 / | // / pi\ \ || | || 0 for |a + --| mod pi = 0| || 2/a 3*pi\ | || \ 4 / | || sec |- - ----| | || | || \2 8 / | || /a 3*pi\ | ||-1 + -------------- | || 2*sec|- - ----| | || 2/a pi\ | || \2 8 / | - |< sec |- + --| | + |<-------------------------------- otherwise | || \2 8 / | ||/ 2/a 3*pi\\ | ||------------------- otherwise | ||| sec |- - ----|| | || 2/a 3*pi\ | ||| \2 8 /| /a pi\ | || sec |- - ----| | |||1 + --------------|*sec|- + --| | || \2 8 / | ||| 2/a pi\ | \2 8 / | || 1 + -------------- | ||| sec |- + --| | | || 2/a pi\ | \\\ \2 8 / / / || sec |- + --| | \\ \2 8 / /
$$\left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{a}{2} - \frac{3 \pi}{8} \right)}}{\left(\frac{\sec^{2}{\left(\frac{a}{2} - \frac{3 \pi}{8} \right)}}{\sec^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}} + 1\right) \sec{\left(\frac{a}{2} + \frac{\pi}{8} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod 2 \pi = 0 \\\frac{\frac{\sec^{2}{\left(\frac{a}{2} - \frac{3 \pi}{8} \right)}}{\sec^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}} - 1}{\frac{\sec^{2}{\left(\frac{a}{2} - \frac{3 \pi}{8} \right)}}{\sec^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}} + 1} & \text{otherwise} \end{cases}\right)$$
// / pi\ \ || 1 for |a + --| mod 2*pi = 0| || \ 4 / | // / pi\ \ || | || 0 for |a + --| mod pi = 0| || 2/a pi\ | || \ 4 / | || cos |- + --| | || | || \2 8 / | || /a pi\ | ||-1 + -------------- | || 2*cos|- + --| | || 2/a 3*pi\ | || \2 8 / | - |< cos |- - ----| | + |<---------------------------------- otherwise | || \2 8 / | ||/ 2/a pi\ \ | ||------------------- otherwise | ||| cos |- + --| | | || 2/a pi\ | ||| \2 8 / | /a 3*pi\ | || cos |- + --| | |||1 + --------------|*cos|- - ----| | || \2 8 / | ||| 2/a 3*pi\| \2 8 / | || 1 + -------------- | ||| cos |- - ----|| | || 2/a 3*pi\ | \\\ \2 8 // / || cos |- - ----| | \\ \2 8 / /
$$\left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{3 \pi}{8} \right)}}\right) \cos{\left(\frac{a}{2} - \frac{3 \pi}{8} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod 2 \pi = 0 \\\frac{-1 + \frac{\cos^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{3 \pi}{8} \right)}}}{1 + \frac{\cos^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{3 \pi}{8} \right)}}} & \text{otherwise} \end{cases}\right)$$
// / pi\ \ || 1 for |a + --| mod 2*pi = 0| || \ 4 / | // / pi\ \ || | || 0 for |a + --| mod pi = 0| || 2/a pi\ | || \ 4 / | || csc |- + --| | || | || \2 8 / | || /a pi\ | ||-1 + ---------------- | || 2*csc|- + --| | || 2/ a 3*pi\ | || \2 8 / | - |< csc |- - + ----| | + |<-------------------------------------- otherwise | || \ 2 8 / | ||/ 2/a pi\ \ | ||--------------------- otherwise | ||| csc |- + --| | | || 2/a pi\ | ||| \2 8 / | / a 3*pi\ | || csc |- + --| | |||1 + ----------------|*csc|- - + ----| | || \2 8 / | ||| 2/ a 3*pi\| \ 2 8 / | || 1 + ---------------- | ||| csc |- - + ----|| | || 2/ a 3*pi\ | \\\ \ 2 8 // / || csc |- - + ----| | \\ \ 2 8 / /
$$\left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}{\left(1 + \frac{\csc^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{3 \pi}{8} \right)}}\right) \csc{\left(- \frac{a}{2} + \frac{3 \pi}{8} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod 2 \pi = 0 \\\frac{-1 + \frac{\csc^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{3 \pi}{8} \right)}}}{1 + \frac{\csc^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{3 \pi}{8} \right)}}} & \text{otherwise} \end{cases}\right)$$
// / pi\ \ // / pi\ \ || 1 for |a + --| mod 2*pi = 0| || 0 for |a + --| mod pi = 0| || \ 4 / | || \ 4 / | || | || | || // /a pi\ \ | || // /a pi\ \ | || || 0 for |- + --| mod pi = 0| | || || 0 for |- + --| mod pi = 0| | - |< || \2 8 / | | + |< || \2 8 / | | ||/ 2/a pi\\ || | | || || | /a pi\ | |||-1 + cot |- + --||*|< / pi\ | otherwise | ||2*|< / pi\ |*cot|- + --| otherwise | ||\ \2 8 // || cos|a + --| | | || || cos|a + --| | \2 8 / | || ||1 \ 4 / | | || ||1 \ 4 / | | || ||- - ----------- otherwise | | || ||- - ----------- otherwise | | \\ \\2 2 / / \\ \\2 2 / /
$$\left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod \pi = 0 \\2 \left(\begin{cases} 0 & \text{for}\: \left(\frac{a}{2} + \frac{\pi}{8}\right) \bmod \pi = 0 \\- \frac{\cos{\left(a + \frac{\pi}{4} \right)}}{2} + \frac{1}{2} & \text{otherwise} \end{cases}\right) \cot{\left(\frac{a}{2} + \frac{\pi}{8} \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} - 1\right) \left(\begin{cases} 0 & \text{for}\: \left(\frac{a}{2} + \frac{\pi}{8}\right) \bmod \pi = 0 \\- \frac{\cos{\left(a + \frac{\pi}{4} \right)}}{2} + \frac{1}{2} & \text{otherwise} \end{cases}\right) & \text{otherwise} \end{cases}\right)$$
// / pi\ \ // / pi\ \ || 1 for |a + --| mod 2*pi = 0| || 0 for |a + --| mod pi = 0| || \ 4 / | || \ 4 / | || | || | || // /a pi\ \ | || // /a pi\ \ | || || 0 for |- + --| mod pi = 0| | || || 0 for |- + --| mod pi = 0| | || || \2 8 / | | || || \2 8 / | | || || | | || || | | - |< || 2/a pi\ | | + |< || 2/a pi\ | | ||/ 2/a pi\\ || 4*cot |- + --| | | || || 4*cot |- + --| | /a pi\ | |||-1 + cot |- + --||*|< \4 16/ | otherwise | ||2*|< \4 16/ |*cot|- + --| otherwise | ||\ \2 8 // ||------------------- otherwise | | || ||------------------- otherwise | \2 8 / | || || 2 | | || || 2 | | || ||/ 2/a pi\\ | | || ||/ 2/a pi\\ | | || |||1 + cot |- + --|| | | || |||1 + cot |- + --|| | | || ||\ \4 16// | | || ||\ \4 16// | | \\ \\ / / \\ \\ / /
$$\left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod \pi = 0 \\2 \left(\begin{cases} 0 & \text{for}\: \left(\frac{a}{2} + \frac{\pi}{8}\right) \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{a}{4} + \frac{\pi}{16} \right)}}{\left(\cot^{2}{\left(\frac{a}{4} + \frac{\pi}{16} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \cot{\left(\frac{a}{2} + \frac{\pi}{8} \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} - 1\right) \left(\begin{cases} 0 & \text{for}\: \left(\frac{a}{2} + \frac{\pi}{8}\right) \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{a}{4} + \frac{\pi}{16} \right)}}{\left(\cot^{2}{\left(\frac{a}{4} + \frac{\pi}{16} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) & \text{otherwise} \end{cases}\right)$$
-Piecewise((1, Mod(a + pi/4 = 2*pi, 0)), ((-1 + cot(a/2 + pi/8)^2)*Piecewise((0, Mod(a/2 + pi/8 = pi, 0)), (4*cot(a/4 + pi/16)^2/(1 + cot(a/4 + pi/16)^2)^2, True)), True)) + Piecewise((0, Mod(a + pi/4 = pi, 0)), (2*Piecewise((0, Mod(a/2 + pi/8 = pi, 0)), (4*cot(a/4 + pi/16)^2/(1 + cot(a/4 + pi/16)^2)^2, True))*cot(a/2 + pi/8), True))