Господин Экзамен
Общий знаменатель ((sin(4*a)/(1+cos(4*a))))*((cos(2*a)/(1+cos(2*a))))
Решение
Вы ввели
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sin(4*a)*cos(2*a) ----------------------------- (1 + cos(4*a))*(1 + cos(2*a))
$$\frac{\sin{\left(4 a \right)} \cos{\left(2 a \right)}}{\left(\cos{\left(2 a \right)} + 1\right) \left(\cos{\left(4 a \right)} + 1\right)}$$
sin(4*a)*cos(2*a)/((1 + cos(4*a))*(1 + cos(2*a)))
Собрать выражение
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sin(2*a) sin(6*a) -------------------------------------- + -------------------------------------- 2 + 2*cos(4*a) + 3*cos(2*a) + cos(6*a) 2 + 2*cos(4*a) + 3*cos(2*a) + cos(6*a)
$$\frac{\sin{\left(2 a \right)}}{3 \cos{\left(2 a \right)} + 2 \cos{\left(4 a \right)} + \cos{\left(6 a \right)} + 2} + \frac{\sin{\left(6 a \right)}}{3 \cos{\left(2 a \right)} + 2 \cos{\left(4 a \right)} + \cos{\left(6 a \right)} + 2}$$
sin(2*a)/(2 + 2*cos(4*a) + 3*cos(2*a) + cos(6*a)) + sin(6*a)/(2 + 2*cos(4*a) + 3*cos(2*a) + cos(6*a))
Рациональный знаменатель
[src]
cos(2*a)*sin(4*a) ------------------------------------------- 1 + cos(2*a)*cos(4*a) + cos(2*a) + cos(4*a)
$$\frac{\sin{\left(4 a \right)} \cos{\left(2 a \right)}}{\cos{\left(2 a \right)} \cos{\left(4 a \right)} + \cos{\left(2 a \right)} + \cos{\left(4 a \right)} + 1}$$
cos(2*a)*sin(4*a)/(1 + cos(2*a)*cos(4*a) + cos(2*a) + cos(4*a))
Численный ответ
[src]
cos(2*a)*sin(4*a)/((1.0 + cos(2*a))*(1.0 + cos(4*a)))
cos(2*a)*sin(4*a)/((1.0 + cos(2*a))*(1.0 + cos(4*a)))
Степени
[src]
/ -2*I*a 2*I*a\
|e e | / -4*I*a 4*I*a\
-I*|------- + ------|*\- e + e /
\ 2 2 /
-----------------------------------------------
/ -4*I*a 4*I*a\ / -2*I*a 2*I*a\
| e e | | e e |
2*|1 + ------- + ------|*|1 + ------- + ------|
\ 2 2 / \ 2 2 /
$$- \frac{i \left(\frac{e^{2 i a}}{2} + \frac{e^{- 2 i a}}{2}\right) \left(e^{4 i a} - e^{- 4 i a}\right)}{2 \left(\frac{e^{2 i a}}{2} + 1 + \frac{e^{- 2 i a}}{2}\right) \left(\frac{e^{4 i a}}{2} + 1 + \frac{e^{- 4 i a}}{2}\right)}$$
-i*(exp(-2*i*a)/2 + exp(2*i*a)/2)*(-exp(-4*i*a) + exp(4*i*a))/(2*(1 + exp(-4*i*a)/2 + exp(4*i*a)/2)*(1 + exp(-2*i*a)/2 + exp(2*i*a)/2))
Общий знаменатель
[src]
cos(2*a)*sin(4*a) ------------------------------------------- 1 + cos(2*a)*cos(4*a) + cos(2*a) + cos(4*a)
$$\frac{\sin{\left(4 a \right)} \cos{\left(2 a \right)}}{\cos{\left(2 a \right)} \cos{\left(4 a \right)} + \cos{\left(2 a \right)} + \cos{\left(4 a \right)} + 1}$$
cos(2*a)*sin(4*a)/(1 + cos(2*a)*cos(4*a) + cos(2*a) + cos(4*a))
Тригонометрическая часть
[src]
tan(a)
$$\tan{\left(a \right)}$$
1 ------ cot(a)
$$\frac{1}{\cot{\left(a \right)}}$$
2 2*sin (a) --------- sin(2*a)
$$\frac{2 \sin^{2}{\left(a \right)}}{\sin{\left(2 a \right)}}$$
sin(2*a) ------------ 1 + cos(2*a)
$$\frac{\sin{\left(2 a \right)}}{\cos{\left(2 a \right)} + 1}$$
sec(a) ----------- / pi\ sec|a - --| \ 2 /
$$\frac{\sec{\left(a \right)}}{\sec{\left(a - \frac{\pi}{2} \right)}}$$
/ pi\ cos|a - --| \ 2 / ----------- cos(a)
$$\frac{\cos{\left(a - \frac{\pi}{2} \right)}}{\cos{\left(a \right)}}$$
/pi \ csc|-- - a| \2 / ----------- csc(a)
$$\frac{\csc{\left(- a + \frac{\pi}{2} \right)}}{\csc{\left(a \right)}}$$
/ 1 \ |1 - ---------|*tan(2*a) | 2 | \ 2*cos (a)/
$$\left(1 - \frac{1}{2 \cos^{2}{\left(a \right)}}\right) \tan{\left(2 a \right)}$$
/ 1 \ / pi\
|1 - ---------|*cos|2*a - --|
| 2 | \ 2 /
\ 2*cos (a)/
-----------------------------
cos(2*a)
$$\frac{\left(1 - \frac{1}{2 \cos^{2}{\left(a \right)}}\right) \cos{\left(2 a - \frac{\pi}{2} \right)}}{\cos{\left(2 a \right)}}$$
/ 2 \
| sec (a)|
|1 - -------|*sec(2*a)
\ 2 /
----------------------
/ pi\
sec|2*a - --|
\ 2 /
$$\frac{\left(- \frac{\sec^{2}{\left(a \right)}}{2} + 1\right) \sec{\left(2 a \right)}}{\sec{\left(2 a - \frac{\pi}{2} \right)}}$$
2 / 1 \
2*sin (2*a)*|1 - --------------|
| 2/ pi\|
| 2*sin |a + --||
\ \ 2 //
--------------------------------
sin(4*a)
$$\frac{2 \cdot \left(1 - \frac{1}{2 \sin^{2}{\left(a + \frac{\pi}{2} \right)}}\right) \sin^{2}{\left(2 a \right)}}{\sin{\left(4 a \right)}}$$
2 / 1 \
cos (2*a)*|2 - -------|*tan(2*a)
| 2 |
\ cos (a)/
--------------------------------
1 + cos(4*a)
$$\frac{\left(2 - \frac{1}{\cos^{2}{\left(a \right)}}\right) \cos^{2}{\left(2 a \right)} \tan{\left(2 a \right)}}{\cos{\left(4 a \right)} + 1}$$
/ 2/pi \\
| csc |-- - a||
| \2 /| /pi \
|1 - ------------|*csc|-- - 2*a|
\ 2 / \2 /
--------------------------------
csc(2*a)
$$\frac{\left(- \frac{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}}{2} + 1\right) \csc{\left(- 2 a + \frac{\pi}{2} \right)}}{\csc{\left(2 a \right)}}$$
/ pi\
cos(2*a)*cos|4*a - --|
\ 2 /
-----------------------------
(1 + cos(2*a))*(1 + cos(4*a))
$$\frac{\cos{\left(2 a \right)} \cos{\left(4 a - \frac{\pi}{2} \right)}}{\left(\cos{\left(2 a \right)} + 1\right) \left(\cos{\left(4 a \right)} + 1\right)}$$
1 ----------------------------------------------- / 1 \ / 1 \ |1 + --------|*|1 + --------|*csc(4*a)*sec(2*a) \ sec(2*a)/ \ sec(4*a)/
$$\frac{1}{\left(1 + \frac{1}{\sec{\left(2 a \right)}}\right) \left(1 + \frac{1}{\sec{\left(4 a \right)}}\right) \csc{\left(4 a \right)} \sec{\left(2 a \right)}}$$
/ 2 \ | / 2/a\\ | | |1 + tan |-|| | | \ \2// | |1 - ----------------|*tan(2*a) | 2| | / 2/a\\ | | 2*|1 - tan |-|| | \ \ \2// /
$$\left(1 - \frac{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}{2 \left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}\right) \tan{\left(2 a \right)}$$
1 ---------------------------------------------------- / 1 \ / 1 \ / pi\ |1 + --------|*|1 + --------|*sec(2*a)*sec|4*a - --| \ sec(2*a)/ \ sec(4*a)/ \ 2 /
$$\frac{1}{\left(1 + \frac{1}{\sec{\left(2 a \right)}}\right) \left(1 + \frac{1}{\sec{\left(4 a \right)}}\right) \sec{\left(2 a \right)} \sec{\left(4 a - \frac{\pi}{2} \right)}}$$
1 ---------------------------------------------------- / 1 \ / 1 \ /pi \ |1 + --------|*|1 + --------|*sec(2*a)*sec|-- - 4*a| \ sec(2*a)/ \ sec(4*a)/ \2 /
$$\frac{1}{\left(1 + \frac{1}{\sec{\left(2 a \right)}}\right) \left(1 + \frac{1}{\sec{\left(4 a \right)}}\right) \sec{\left(2 a \right)} \sec{\left(- 4 a + \frac{\pi}{2} \right)}}$$
/ / 1 for a mod 2*pi = 0\ | | | | < 2 | | |------------ otherwise | | \1 + cos(2*a) | |1 - ---------------------------------|*tan(2*a) \ 2 /
$$\left(\left(- \frac{\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{2}{\cos{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}}{2}\right) + 1\right) \tan{\left(2 a \right)}$$
/pi \
sin(4*a)*sin|-- + 2*a|
\2 /
---------------------------------------
/ /pi \\ / /pi \\
|1 + sin|-- + 2*a||*|1 + sin|-- + 4*a||
\ \2 // \ \2 //
$$\frac{\sin{\left(4 a \right)} \sin{\left(2 a + \frac{\pi}{2} \right)}}{\left(\sin{\left(2 a + \frac{\pi}{2} \right)} + 1\right) \left(\sin{\left(4 a + \frac{\pi}{2} \right)} + 1\right)}$$
2/ pi\ / 2/ pi\\ / 2 \
sin |2*a + --|*|1 - cot |2*a + --||*\1 - tan (a)/
\ 4 / \ \ 4 //
-------------------------------------------------
2*(1 + cos(4*a))
$$\frac{\left(- \tan^{2}{\left(a \right)} + 1\right) \left(- \cot^{2}{\left(2 a + \frac{\pi}{4} \right)} + 1\right) \sin^{2}{\left(2 a + \frac{\pi}{4} \right)}}{2 \left(\cos{\left(4 a \right)} + 1\right)}$$
1 -------------------------------------------------------------- / 1 \ / 1 \ /pi \ |1 + -------------|*|1 + -------------|*csc(4*a)*csc|-- - 2*a| | /pi \| | /pi \| \2 / | csc|-- - 4*a|| | csc|-- - 2*a|| \ \2 // \ \2 //
$$\frac{1}{\left(1 + \frac{1}{\csc{\left(- 4 a + \frac{\pi}{2} \right)}}\right) \left(1 + \frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}}\right) \csc{\left(4 a \right)} \csc{\left(- 2 a + \frac{\pi}{2} \right)}}$$
1 ------------------------------------------------------------------- / 1 \ / 1 \ /pi \ |1 + -------------|*|1 + -------------|*csc(pi - 4*a)*csc|-- - 2*a| | /pi \| | /pi \| \2 / | csc|-- - 4*a|| | csc|-- - 2*a|| \ \2 // \ \2 //
$$\frac{1}{\left(1 + \frac{1}{\csc{\left(- 4 a + \frac{\pi}{2} \right)}}\right) \left(1 + \frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}}\right) \csc{\left(- 4 a + \pi \right)} \csc{\left(- 2 a + \frac{\pi}{2} \right)}}$$
2 / pi\
2*cos (2*a)*tan(2*a)*tan|a + --|
\ 4 /
-----------------------------------------
/ 2/ pi\\ 2
|1 + tan |a + --||*(1 + cos(4*a))*cos (a)
\ \ 4 //
$$\frac{2 \cos^{2}{\left(2 a \right)} \tan{\left(2 a \right)} \tan{\left(a + \frac{\pi}{4} \right)}}{\left(\cos{\left(4 a \right)} + 1\right) \left(\tan^{2}{\left(a + \frac{\pi}{4} \right)} + 1\right) \cos^{2}{\left(a \right)}}$$
2 / pi\
2*sin (2*a)*cot(2*a)*tan|a + --|
\ 4 /
-----------------------------------------
/ 2/ pi\\ 2
|1 + tan |a + --||*(1 + cos(4*a))*cos (a)
\ \ 4 //
$$\frac{2 \sin^{2}{\left(2 a \right)} \tan{\left(a + \frac{\pi}{4} \right)} \cot{\left(2 a \right)}}{\left(\cos{\left(4 a \right)} + 1\right) \left(\tan^{2}{\left(a + \frac{\pi}{4} \right)} + 1\right) \cos^{2}{\left(a \right)}}$$
/ 1 for a mod 2*pi = 0
|
| 2
| / 2/a\\
| |1 + cot |-||
< \ \2//
|--------------- otherwise
| 2
|/ 2/a\\
||-1 + cot |-||
\\ \2//
1 - ------------------------------------
2
----------------------------------------
cot(2*a)
$$\frac{\left(- \frac{\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{2}} & \text{otherwise} \end{cases}}{2}\right) + 1}{\cot{\left(2 a \right)}}$$
2
/ 2 2 \
4*\cos (a) - sin (a)/ *cos(a)*sin(a)
----------------------------------------------------------------------
/ 2 \
| / 2 2 \ 2 2 | / 2 2 \
\1 + \cos (a) - sin (a)/ - 4*cos (a)*sin (a)/*\1 + cos (a) - sin (a)/
$$\frac{4 \left(- \sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}\right)^{2} \sin{\left(a \right)} \cos{\left(a \right)}}{\left(- 4 \sin^{2}{\left(a \right)} \cos^{2}{\left(a \right)} + \left(- \sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}\right)^{2} + 1\right) \left(- \sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)} + 1\right)}$$
/ 2 \
2*\1 - tan (a)/*tan(2*a)
-------------------------------------------------------------------
/ 2 \ / 2 \
/ 2 \ / 2 \ | 1 - tan (a)| | 1 - tan (2*a)|
\1 + tan (a)/*\1 + tan (2*a)/*|1 + -----------|*|1 + -------------|
| 2 | | 2 |
\ 1 + tan (a)/ \ 1 + tan (2*a)/
$$\frac{2 \cdot \left(- \tan^{2}{\left(a \right)} + 1\right) \tan{\left(2 a \right)}}{\left(\frac{- \tan^{2}{\left(a \right)} + 1}{\tan^{2}{\left(a \right)} + 1} + 1\right) \left(\frac{- \tan^{2}{\left(2 a \right)} + 1}{\tan^{2}{\left(2 a \right)} + 1} + 1\right) \left(\tan^{2}{\left(a \right)} + 1\right) \left(\tan^{2}{\left(2 a \right)} + 1\right)}$$
/ 2 \ / 2/ pi\\
\-1 + cot (a)/*|-1 + tan |2*a + --||
\ \ 4 //
--------------------------------------------------------------------------
/ 2 \ / 2 \
/ 2 \ / 2/ pi\\ | -1 + cot (a)| | -1 + cot (2*a)|
\1 + cot (a)/*|1 + tan |2*a + --||*|1 + ------------|*|1 + --------------|
\ \ 4 // | 2 | | 2 |
\ 1 + cot (a) / \ 1 + cot (2*a) /
$$\frac{\left(\tan^{2}{\left(2 a + \frac{\pi}{4} \right)} - 1\right) \left(\cot^{2}{\left(a \right)} - 1\right)}{\left(\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} + 1\right) \left(\frac{\cot^{2}{\left(2 a \right)} - 1}{\cot^{2}{\left(2 a \right)} + 1} + 1\right) \left(\tan^{2}{\left(2 a + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(a \right)} + 1\right)}$$
/ 2/ pi\\ / 2 \
|1 - cot |2*a + --||*\1 - tan (a)/
\ \ 4 //
------------------------------------------------------------------------
/ 2 \ / 2 \
/ 2/ pi\\ / 2 \ | 1 - tan (a)| | 1 - tan (2*a)|
|1 + cot |2*a + --||*\1 + tan (a)/*|1 + -----------|*|1 + -------------|
\ \ 4 // | 2 | | 2 |
\ 1 + tan (a)/ \ 1 + tan (2*a)/
$$\frac{\left(- \tan^{2}{\left(a \right)} + 1\right) \left(- \cot^{2}{\left(2 a + \frac{\pi}{4} \right)} + 1\right)}{\left(\frac{- \tan^{2}{\left(a \right)} + 1}{\tan^{2}{\left(a \right)} + 1} + 1\right) \left(\frac{- \tan^{2}{\left(2 a \right)} + 1}{\tan^{2}{\left(2 a \right)} + 1} + 1\right) \left(\tan^{2}{\left(a \right)} + 1\right) \left(\cot^{2}{\left(2 a + \frac{\pi}{4} \right)} + 1\right)}$$
/ pi\
4*tan(2*a)*tan|a + --|
\ 4 /
----------------------------------------------------------------------------------
/ / pi\ \ / / pi\ \
| 2*tan|a + --| | | 2*tan|2*a + --| |
/ 2 \ / 2/ pi\\ | \ 4 / | | \ 4 / |
\1 + tan (2*a)/*|1 + tan |a + --||*|1 + ----------------|*|1 + ------------------|
\ \ 4 // | 2/ pi\| | 2/ pi\|
| 1 + tan |a + --|| | 1 + tan |2*a + --||
\ \ 4 // \ \ 4 //
$$\frac{4 \tan{\left(2 a \right)} \tan{\left(a + \frac{\pi}{4} \right)}}{\left(1 + \frac{2 \tan{\left(a + \frac{\pi}{4} \right)}}{\tan^{2}{\left(a + \frac{\pi}{4} \right)} + 1}\right) \left(1 + \frac{2 \tan{\left(2 a + \frac{\pi}{4} \right)}}{\tan^{2}{\left(2 a + \frac{\pi}{4} \right)} + 1}\right) \left(\tan^{2}{\left(2 a \right)} + 1\right) \left(\tan^{2}{\left(a + \frac{\pi}{4} \right)} + 1\right)}$$
/ pi\
4*cot(2*a)*tan|a + --|
\ 4 /
----------------------------------------------------------------------------------
/ / pi\ \ / / pi\ \
| 2*tan|a + --| | | 2*tan|2*a + --| |
/ 2 \ / 2/ pi\\ | \ 4 / | | \ 4 / |
\1 + cot (2*a)/*|1 + tan |a + --||*|1 + ----------------|*|1 + ------------------|
\ \ 4 // | 2/ pi\| | 2/ pi\|
| 1 + tan |a + --|| | 1 + tan |2*a + --||
\ \ 4 // \ \ 4 //
$$\frac{4 \tan{\left(a + \frac{\pi}{4} \right)} \cot{\left(2 a \right)}}{\left(1 + \frac{2 \tan{\left(a + \frac{\pi}{4} \right)}}{\tan^{2}{\left(a + \frac{\pi}{4} \right)} + 1}\right) \left(1 + \frac{2 \tan{\left(2 a + \frac{\pi}{4} \right)}}{\tan^{2}{\left(2 a + \frac{\pi}{4} \right)} + 1}\right) \left(\tan^{2}{\left(a + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(2 a \right)} + 1\right)}$$
/ 1 \
2*|1 - -------|
| 2 |
\ cot (a)/
----------------------------------------------------------------------------
/ 1 \ / 1 \
| 1 - -------| | 1 - ---------|
| 2 | | 2 |
/ 1 \ / 1 \ | cot (a)| | cot (2*a)|
|1 + -------|*|1 + ---------|*|1 + -----------|*|1 + -------------|*cot(2*a)
| 2 | | 2 | | 1 | | 1 |
\ cot (a)/ \ cot (2*a)/ | 1 + -------| | 1 + ---------|
| 2 | | 2 |
\ cot (a)/ \ cot (2*a)/
$$\frac{2 \cdot \left(1 - \frac{1}{\cot^{2}{\left(a \right)}}\right)}{\left(1 + \frac{1}{\cot^{2}{\left(a \right)}}\right) \left(1 + \frac{1}{\cot^{2}{\left(2 a \right)}}\right) \left(\frac{1 - \frac{1}{\cot^{2}{\left(a \right)}}}{1 + \frac{1}{\cot^{2}{\left(a \right)}}} + 1\right) \left(\frac{1 - \frac{1}{\cot^{2}{\left(2 a \right)}}}{1 + \frac{1}{\cot^{2}{\left(2 a \right)}}} + 1\right) \cot{\left(2 a \right)}}$$
// 0 for 4*a mod pi = 0\ // 1 for a mod pi = 0\
|< |*|< |
\\sin(4*a) otherwise / \\cos(2*a) otherwise /
-------------------------------------------------------------------------------------------
/ // 1 for 2*a mod pi = 0\\
/ // 1 for a mod pi = 0\\ | || ||
|1 + |< ||*|1 + |< 2 / 2 \ ||
\ \\cos(2*a) otherwise // | ||sin (2*a)*\-1 + cot (2*a)/ otherwise ||
\ \\ //
$$\frac{\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right)}{\left(\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right) + 1\right) \left(\left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\left(\cot^{2}{\left(2 a \right)} - 1\right) \sin^{2}{\left(2 a \right)} & \text{otherwise} \end{cases}\right) + 1\right)}$$
// 0 for 4*a mod pi = 0\ // 1 for a mod pi = 0\
|< |*|< |
\\sin(4*a) otherwise / \\cos(2*a) otherwise /
-------------------------------------------------------------------------------
/ // 1 for 2*a mod pi = 0\\
| || ||
| || 1 ||
/ // 1 for a mod pi = 0\\ | ||-1 + --------- ||
|1 + |< ||*|1 + |< 2 ||
\ \\cos(2*a) otherwise // | || tan (2*a) ||
| ||-------------- otherwise ||
| || 2 ||
\ \\ csc (2*a) //
$$\frac{\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right)}{\left(\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right) + 1\right) \left(\left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(2 a \right)}}}{\csc^{2}{\left(2 a \right)}} & \text{otherwise} \end{cases}\right) + 1\right)}$$
// 0 for 4*a mod pi = 0\ // 1 for a mod pi = 0\
|< |*|< |
\\sin(4*a) otherwise / \\cos(2*a) otherwise /
-------------------------------------------------------------------------------------------
/ // 1 for 2*a mod pi = 0\\
| || ||
/ // 1 for a mod pi = 0\\ | || 2 / 1 \ ||
|1 + |< ||*|1 + |
$$\frac{\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right)}{\left(\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right) + 1\right) \left(\left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\left(-1 + \frac{1}{\tan^{2}{\left(2 a \right)}}\right) \sin^{2}{\left(2 a \right)} & \text{otherwise} \end{cases}\right) + 1\right)}$$
// / 3*pi\ \
// 1 for a mod pi = 0\ || 1 for |4*a + ----| mod 2*pi = 0|
|< |*|< \ 2 / |
\\cos(2*a) otherwise / || |
\\sin(4*a) otherwise /
-------------------------------------------------------------------------------------------
/ // 1 for 2*a mod pi = 0\\
/ // 1 for a mod pi = 0\\ | || ||
|1 + |< ||*|1 + |< 2 / 2 \ ||
\ \\cos(2*a) otherwise // | ||sin (2*a)*\-1 + cot (2*a)/ otherwise ||
\ \\ //
$$\frac{\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(4 a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right)}{\left(\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right) + 1\right) \left(\left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\left(\cot^{2}{\left(2 a \right)} - 1\right) \sin^{2}{\left(2 a \right)} & \text{otherwise} \end{cases}\right) + 1\right)}$$
// 0 for 4*a mod pi = 0\ // 1 for a mod pi = 0\
|< |*|< |
\\sin(4*a) otherwise / \\cos(2*a) otherwise /
---------------------------------------------------------------------------------------------------
/ // 1 for 2*a mod pi = 0\\
| || ||
/ // 1 for a mod pi = 0\\ | || 4 2 / 1 \ ||
|1 + |< ||*|1 + |<4*cos (a)*tan (a)*|-1 + ---------| otherwise ||
\ \\cos(2*a) otherwise // | || | 2 | ||
| || \ tan (2*a)/ ||
\ \\ //
$$\frac{\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right)}{\left(\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right) + 1\right) \left(\left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\4 \left(-1 + \frac{1}{\tan^{2}{\left(2 a \right)}}\right) \cos^{4}{\left(a \right)} \tan^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1\right)}$$
// 0 for 4*a mod pi = 0\ // 1 for a mod pi = 0\
|< |*|< |
\\sin(4*a) otherwise / \\cos(2*a) otherwise /
---------------------------------------------------------------------------------------------
/ // 1 for 2*a mod pi = 0\\
| || ||
/ // 1 for a mod pi = 0\\ | || / 2 \ ||
|1 + |< ||*|1 + |< 2 | sin (4*a) | ||
\ \\cos(2*a) otherwise // | ||sin (2*a)*|-1 + -----------| otherwise ||
| || | 4 | ||
\ \\ \ 4*sin (2*a)/ //
$$\frac{\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right)}{\left(\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right) + 1\right) \left(\left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\left(-1 + \frac{\sin^{2}{\left(4 a \right)}}{4 \sin^{4}{\left(2 a \right)}}\right) \sin^{2}{\left(2 a \right)} & \text{otherwise} \end{cases}\right) + 1\right)}$$
// /pi \ \
// 0 for 4*a mod pi = 0\ || 0 for |-- + 2*a| mod pi = 0|
|< |*|< \2 / |
\\sin(4*a) otherwise / || |
\\cos(2*a) otherwise /
-----------------------------------------------------------------------------------------
/ // /pi \ \\ / // /pi \ \\
| || 0 for |-- + 2*a| mod pi = 0|| | || 0 for |-- + 4*a| mod pi = 0||
|1 + |< \2 / ||*|1 + |< \2 / ||
| || || | || ||
\ \\cos(2*a) otherwise // \ \\cos(4*a) otherwise //
$$\frac{\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(2 a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right)}{\left(\left(\begin{cases} 0 & \text{for}\: \left(2 a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right) + 1\right) \left(\left(\begin{cases} 0 & \text{for}\: \left(4 a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\cos{\left(4 a \right)} & \text{otherwise} \end{cases}\right) + 1\right)}$$
// 0 for 4*a mod pi = 0\ // 1 for a mod pi = 0\
|< |*|< |
\\sin(4*a) otherwise / \\cos(2*a) otherwise /
--------------------------------------------------------------------------------------------------
/ // 1 for 2*a mod pi = 0\\
| || ||
| || / 2 \ ||
/ // 1 for a mod pi = 0\\ | || | sin (4*a) | ||
|1 + |< ||*|1 + |<(1 - cos(4*a))*|-1 + -----------| ||
\ \\cos(2*a) otherwise // | || | 4 | ||
| || \ 4*sin (2*a)/ ||
| ||--------------------------------- otherwise ||
\ \\ 2 //
$$\frac{\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right)}{\left(\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right) + 1\right) \left(\left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{\left(-1 + \frac{\sin^{2}{\left(4 a \right)}}{4 \sin^{4}{\left(2 a \right)}}\right) \left(- \cos{\left(4 a \right)} + 1\right)}{2} & \text{otherwise} \end{cases}\right) + 1\right)}$$
// 1 for a mod pi = 0\
// 0 for 4*a mod pi = 0\ || |
|< |*|< /pi \ |
\\sin(4*a) otherwise / ||sin|-- + 2*a| otherwise |
\\ \2 / /
--------------------------------------------------------------------------------------------------
/ // 1 for 2*a mod pi = 0\\
/ // 1 for a mod pi = 0\\ | || ||
| || || | || / 2 \ ||
|1 + |< /pi \ ||*|1 + |< 2 | sin (4*a) | ||
| ||sin|-- + 2*a| otherwise || | ||sin (2*a)*|-1 + -----------| otherwise ||
\ \\ \2 / // | || | 4 | ||
\ \\ \ 4*sin (2*a)/ //
$$\frac{\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(2 a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)}{\left(\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(2 a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + 1\right) \left(\left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\left(-1 + \frac{\sin^{2}{\left(4 a \right)}}{4 \sin^{4}{\left(2 a \right)}}\right) \sin^{2}{\left(2 a \right)} & \text{otherwise} \end{cases}\right) + 1\right)}$$
/ 4 \
2 | 4*sin (a)|
4*sin (2*a)*|1 - ---------|
| 2 |
\ sin (2*a)/
------------------------------------------------------------------------------------
/ 4 \ / 4 \
| 4*sin (a)| | 4*sin (2*a)|
| 1 - ---------| | 1 - -----------|
| 2 | | 2 | / 4 \ / 4 \
| sin (2*a)| | sin (4*a) | | 4*sin (a)| | 4*sin (2*a)|
|1 + -------------|*|1 + ---------------|*|1 + ---------|*|1 + -----------|*sin(4*a)
| 4 | | 4 | | 2 | | 2 |
| 4*sin (a)| | 4*sin (2*a)| \ sin (2*a)/ \ sin (4*a) /
| 1 + ---------| | 1 + -----------|
| 2 | | 2 |
\ sin (2*a)/ \ sin (4*a) /
$$\frac{4 \left(- \frac{4 \sin^{4}{\left(a \right)}}{\sin^{2}{\left(2 a \right)}} + 1\right) \sin^{2}{\left(2 a \right)}}{\left(\frac{- \frac{4 \sin^{4}{\left(a \right)}}{\sin^{2}{\left(2 a \right)}} + 1}{\frac{4 \sin^{4}{\left(a \right)}}{\sin^{2}{\left(2 a \right)}} + 1} + 1\right) \left(\frac{- \frac{4 \sin^{4}{\left(2 a \right)}}{\sin^{2}{\left(4 a \right)}} + 1}{\frac{4 \sin^{4}{\left(2 a \right)}}{\sin^{2}{\left(4 a \right)}} + 1} + 1\right) \left(\frac{4 \sin^{4}{\left(a \right)}}{\sin^{2}{\left(2 a \right)}} + 1\right) \left(\frac{4 \sin^{4}{\left(2 a \right)}}{\sin^{2}{\left(4 a \right)}} + 1\right) \sin{\left(4 a \right)}}$$
// 0 for 4*a mod pi = 0\
|| | // 1 for a mod pi = 0\
|< / pi\ |*|< |
||cos|4*a - --| otherwise | \\cos(2*a) otherwise /
\\ \ 2 / /
-----------------------------------------------------------------------------------------------------
/ // 1 for 2*a mod pi = 0\\
| || ||
| || / 2 \ ||
/ // 1 for a mod pi = 0\\ | || 2/ pi\ | cos (2*a) | ||
|1 + |< ||*|1 + |
$$\frac{\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\cos{\left(4 a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right)}{\left(\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right) + 1\right) \left(\left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\left(\frac{\cos^{2}{\left(2 a \right)}}{\cos^{2}{\left(2 a - \frac{\pi}{2} \right)}} - 1\right) \cos^{2}{\left(2 a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + 1\right)}$$
// 0 for 4*a mod pi = 0\
|| | // 1 for a mod pi = 0\
|| 1 | || |
|<------------- otherwise |*|< 1 |
|| / pi\ | ||-------- otherwise |
||sec|4*a - --| | \\sec(2*a) /
\\ \ 2 / /
------------------------------------------------------------------------------------
/ // 1 for 2*a mod pi = 0\\
| || ||
| || 2/ pi\ ||
| || sec |2*a - --| ||
/ // 1 for a mod pi = 0\\ | || \ 2 / ||
| || || | ||-1 + -------------- ||
|1 + |< 1 ||*|1 + |< 2 ||
| ||-------- otherwise || | || sec (2*a) ||
\ \\sec(2*a) // | ||------------------- otherwise ||
| || 2/ pi\ ||
| || sec |2*a - --| ||
| || \ 2 / ||
\ \\ //
$$\frac{\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{1}{\sec{\left(4 a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sec{\left(2 a \right)}} & \text{otherwise} \end{cases}\right)}{\left(\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sec{\left(2 a \right)}} & \text{otherwise} \end{cases}\right) + 1\right) \left(\left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(2 a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(2 a \right)}}}{\sec^{2}{\left(2 a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1\right)}$$
// 1 for a mod pi = 0\
// 0 for 4*a mod pi = 0\ || |
|| | || 1 |
|< 1 |*|<------------- otherwise |
||-------- otherwise | || /pi \ |
\\csc(4*a) / ||csc|-- - 2*a| |
\\ \2 / /
-----------------------------------------------------------------------------------------
/ // 1 for 2*a mod pi = 0\\
| || ||
/ // 1 for a mod pi = 0\\ | || 2 ||
| || || | || csc (2*a) ||
| || 1 || | ||-1 + -------------- ||
|1 + |<------------- otherwise ||*|1 + |< 2/pi \ ||
| || /pi \ || | || csc |-- - 2*a| ||
| ||csc|-- - 2*a| || | || \2 / ||
\ \\ \2 / // | ||------------------- otherwise ||
| || 2 ||
\ \\ csc (2*a) //
$$\frac{\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{1}{\csc{\left(4 a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)}{\left(\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1\right) \left(\left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{\frac{\csc^{2}{\left(2 a \right)}}{\csc^{2}{\left(- 2 a + \frac{\pi}{2} \right)}} - 1}{\csc^{2}{\left(2 a \right)}} & \text{otherwise} \end{cases}\right) + 1\right)}$$
// 0 for 4*a mod pi = 0\ // 1 for a mod pi = 0\
|| | || |
|| 2*cot(2*a) | || 2 |
|<------------- otherwise |*|<-1 + cot (a) |
|| 2 | ||------------ otherwise |
||1 + cot (2*a) | || 2 |
\\ / \\1 + cot (a) /
-----------------------------------------------------------------------------------
/ // 1 for a mod pi = 0\\ / // 1 for 2*a mod pi = 0\\
| || || | || ||
| || 2 || | || 2 ||
|1 + |<-1 + cot (a) ||*|1 + |<-1 + cot (2*a) ||
| ||------------ otherwise || | ||-------------- otherwise ||
| || 2 || | || 2 ||
\ \\1 + cot (a) // \ \\1 + cot (2*a) //
$$\frac{\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{2 \cot{\left(2 a \right)}}{\cot^{2}{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right)}{\left(\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right) + 1\right) \left(\left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{\cot^{2}{\left(2 a \right)} - 1}{\cot^{2}{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}\right) + 1\right)}$$
/ 2/ pi\\
| cos |a - --||
| \ 2 /| / pi\
2*|1 - ------------|*cos|2*a - --|
| 2 | \ 2 /
\ cos (a) /
------------------------------------------------------------------------------------------------
/ 2/ pi\\ / 2/ pi\\
| cos |a - --|| | cos |2*a - --||
| \ 2 /| | \ 2 /|
| 1 - ------------| | 1 - --------------| / 2/ pi\\ / 2/ pi\\
| 2 | | 2 | | cos |a - --|| | cos |2*a - --||
| cos (a) | | cos (2*a) | | \ 2 /| | \ 2 /|
|1 + ----------------|*|1 + ------------------|*|1 + ------------|*|1 + --------------|*cos(2*a)
| 2/ pi\| | 2/ pi\| | 2 | | 2 |
| cos |a - --|| | cos |2*a - --|| \ cos (a) / \ cos (2*a) /
| \ 2 /| | \ 2 /|
| 1 + ------------| | 1 + --------------|
| 2 | | 2 |
\ cos (a) / \ cos (2*a) /
$$\frac{2 \cdot \left(1 - \frac{\cos^{2}{\left(a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(a \right)}}\right) \cos{\left(2 a - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(a \right)}}\right) \left(1 + \frac{\cos^{2}{\left(2 a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(2 a \right)}}\right) \left(\frac{1 - \frac{\cos^{2}{\left(a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(a \right)}}}{1 + \frac{\cos^{2}{\left(a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(a \right)}}} + 1\right) \left(\frac{1 - \frac{\cos^{2}{\left(2 a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(2 a \right)}}}{1 + \frac{\cos^{2}{\left(2 a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(2 a \right)}}} + 1\right) \cos{\left(2 a \right)}}$$
/ 2 \
| sec (a) |
2*|1 - ------------|*sec(2*a)
| 2/ pi\|
| sec |a - --||
\ \ 2 //
-----------------------------------------------------------------------------------------------------
/ 2 \ / 2 \
| sec (a) | | sec (2*a) |
| 1 - ------------| | 1 - --------------|
| 2/ pi\| | 2/ pi\|
| sec |a - --|| | sec |2*a - --|| / 2 \ / 2 \
| \ 2 /| | \ 2 /| | sec (a) | | sec (2*a) | / pi\
|1 + ----------------|*|1 + ------------------|*|1 + ------------|*|1 + --------------|*sec|2*a - --|
| 2 | | 2 | | 2/ pi\| | 2/ pi\| \ 2 /
| sec (a) | | sec (2*a) | | sec |a - --|| | sec |2*a - --||
| 1 + ------------| | 1 + --------------| \ \ 2 // \ \ 2 //
| 2/ pi\| | 2/ pi\|
| sec |a - --|| | sec |2*a - --||
\ \ 2 // \ \ 2 //
$$\frac{2 \left(- \frac{\sec^{2}{\left(a \right)}}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(2 a \right)}}{\left(\frac{- \frac{\sec^{2}{\left(a \right)}}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} + 1}{\frac{\sec^{2}{\left(a \right)}}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} + 1} + 1\right) \left(\frac{- \frac{\sec^{2}{\left(2 a \right)}}{\sec^{2}{\left(2 a - \frac{\pi}{2} \right)}} + 1}{\frac{\sec^{2}{\left(2 a \right)}}{\sec^{2}{\left(2 a - \frac{\pi}{2} \right)}} + 1} + 1\right) \left(\frac{\sec^{2}{\left(a \right)}}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} + 1\right) \left(\frac{\sec^{2}{\left(2 a \right)}}{\sec^{2}{\left(2 a - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(2 a - \frac{\pi}{2} \right)}}$$
/ 2/pi \\
| csc |-- - a||
| \2 /| /pi \
2*|1 - ------------|*csc|-- - 2*a|
| 2 | \2 /
\ csc (a) /
------------------------------------------------------------------------------------------------
/ 2/pi \\ / 2/pi \\
| csc |-- - a|| | csc |-- - 2*a||
| \2 /| | \2 /|
| 1 - ------------| | 1 - --------------| / 2/pi \\ / 2/pi \\
| 2 | | 2 | | csc |-- - a|| | csc |-- - 2*a||
| csc (a) | | csc (2*a) | | \2 /| | \2 /|
|1 + ----------------|*|1 + ------------------|*|1 + ------------|*|1 + --------------|*csc(2*a)
| 2/pi \| | 2/pi \| | 2 | | 2 |
| csc |-- - a|| | csc |-- - 2*a|| \ csc (a) / \ csc (2*a) /
| \2 /| | \2 /|
| 1 + ------------| | 1 + --------------|
| 2 | | 2 |
\ csc (a) / \ csc (2*a) /
$$\frac{2 \cdot \left(1 - \frac{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(a \right)}}\right) \csc{\left(- 2 a + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(a \right)}}\right) \left(1 + \frac{\csc^{2}{\left(- 2 a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(2 a \right)}}\right) \left(\frac{1 - \frac{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(a \right)}}}{1 + \frac{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(a \right)}}} + 1\right) \left(\frac{1 - \frac{\csc^{2}{\left(- 2 a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(2 a \right)}}}{1 + \frac{\csc^{2}{\left(- 2 a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(2 a \right)}}} + 1\right) \csc{\left(2 a \right)}}$$
// / 3*pi\ \
|| 1 for |4*a + ----| mod 2*pi = 0|
// 1 for a mod pi = 0\ || \ 2 / |
|| | || |
|| 2 | || 2/ pi\ |
|<-1 + cot (a) |*|<-1 + tan |2*a + --| |
||------------ otherwise | || \ 4 / |
|| 2 | ||------------------- otherwise |
\\1 + cot (a) / || 2/ pi\ |
|| 1 + tan |2*a + --| |
\\ \ 4 / /
---------------------------------------------------------------------------------------
/ // 1 for a mod pi = 0\\ / // 1 for 2*a mod pi = 0\\
| || || | || ||
| || 2 || | || 2 ||
|1 + |<-1 + cot (a) ||*|1 + |<-1 + cot (2*a) ||
| ||------------ otherwise || | ||-------------- otherwise ||
| || 2 || | || 2 ||
\ \\1 + cot (a) // \ \\1 + cot (2*a) //
$$\frac{\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(4 a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(2 a + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(2 a + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)}{\left(\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right) + 1\right) \left(\left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{\cot^{2}{\left(2 a \right)} - 1}{\cot^{2}{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}\right) + 1\right)}$$
// 0 for 4*a mod pi = 0\ // 1 for a mod pi = 0\
|| | || |
|| 2*tan(2*a) | || 2 |
|<------------- otherwise |*|<1 - tan (a) |
|| 2 | ||----------- otherwise |
||1 + tan (2*a) | || 2 |
\\ / \\1 + tan (a) /
----------------------------------------------------------------------------------------------
/ // 1 for 2*a mod pi = 0\\
| || ||
/ // 1 for a mod pi = 0\\ | || 2 / 1 \ ||
| || || | ||4*tan (a)*|-1 + ---------| ||
| || 2 || | || | 2 | ||
|1 + |<1 - tan (a) ||*|1 + |< \ tan (2*a)/ ||
| ||----------- otherwise || | ||-------------------------- otherwise ||
| || 2 || | || 2 ||
\ \\1 + tan (a) // | || / 2 \ ||
| || \1 + tan (a)/ ||
\ \\ //
$$\frac{\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{2 \tan{\left(2 a \right)}}{\tan^{2}{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{- \tan^{2}{\left(a \right)} + 1}{\tan^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right)}{\left(\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{- \tan^{2}{\left(a \right)} + 1}{\tan^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right) + 1\right) \left(\left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{4 \left(-1 + \frac{1}{\tan^{2}{\left(2 a \right)}}\right) \tan^{2}{\left(a \right)}}{\left(\tan^{2}{\left(a \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) + 1\right)}$$
// 1 for a mod pi = 0\
// 0 for 4*a mod pi = 0\ || |
|| | || 1 |
|| 2 | ||-1 + ------- |
||------------------------ otherwise | || 2 |
|
$$\frac{\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(2 a \right)}}\right) \tan{\left(2 a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(a \right)}}}{1 + \frac{1}{\tan^{2}{\left(a \right)}}} & \text{otherwise} \end{cases}\right)}{\left(\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(a \right)}}}{1 + \frac{1}{\tan^{2}{\left(a \right)}}} & \text{otherwise} \end{cases}\right) + 1\right) \left(\left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(2 a \right)}}}{1 + \frac{1}{\tan^{2}{\left(2 a \right)}}} & \text{otherwise} \end{cases}\right) + 1\right)}$$
// /pi \ \
|| 0 for |-- + 2*a| mod pi = 0|
// 0 for 4*a mod pi = 0\ || \2 / |
|| | || |
|| 2*cot(2*a) | || / pi\ |
|<------------- otherwise |*|< 2*cot|a + --| |
|| 2 | || \ 4 / |
||1 + cot (2*a) | ||---------------- otherwise |
\\ / || 2/ pi\ |
||1 + cot |a + --| |
\\ \ 4 / /
-----------------------------------------------------------------------------------------------------------
/ // /pi \ \\ / // /pi \ \\
| || 0 for |-- + 2*a| mod pi = 0|| | || 0 for |-- + 4*a| mod pi = 0||
| || \2 / || | || \2 / ||
| || || | || ||
| || / pi\ || | || / pi\ ||
|1 + |< 2*cot|a + --| ||*|1 + |< 2*cot|2*a + --| ||
| || \ 4 / || | || \ 4 / ||
| ||---------------- otherwise || | ||------------------ otherwise ||
| || 2/ pi\ || | || 2/ pi\ ||
| ||1 + cot |a + --| || | ||1 + cot |2*a + --| ||
\ \\ \ 4 / // \ \\ \ 4 / //
$$\frac{\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{2 \cot{\left(2 a \right)}}{\cot^{2}{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(2 a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(a + \frac{\pi}{4} \right)}}{\cot^{2}{\left(a + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)}{\left(\left(\begin{cases} 0 & \text{for}\: \left(2 a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(a + \frac{\pi}{4} \right)}}{\cot^{2}{\left(a + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1\right) \left(\left(\begin{cases} 0 & \text{for}\: \left(4 a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(2 a + \frac{\pi}{4} \right)}}{\cot^{2}{\left(2 a + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1\right)}$$
// 0 for 4*a mod pi = 0\ // 1 for a mod pi = 0\
|| | || |
|
$$\frac{\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)}{\left(\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + 1\right) \left(\left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\left(\cot^{2}{\left(2 a \right)} - 1\right) \left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{- \cos{\left(4 a \right)} + 1}{2} & \text{otherwise} \end{cases}\right) & \text{otherwise} \end{cases}\right) + 1\right)}$$
// 1 for a mod pi = 0\
|| |
// 0 for 4*a mod pi = 0\ || 2 |
|| | || sin (2*a) |
|| sin(4*a) | ||-1 + --------- |
||--------------------------- otherwise | || 4 |
|
$$\frac{\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{\sin{\left(4 a \right)}}{\left(1 + \frac{\sin^{2}{\left(4 a \right)}}{4 \sin^{4}{\left(2 a \right)}}\right) \sin^{2}{\left(2 a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(2 a \right)}}{4 \sin^{4}{\left(a \right)}}}{1 + \frac{\sin^{2}{\left(2 a \right)}}{4 \sin^{4}{\left(a \right)}}} & \text{otherwise} \end{cases}\right)}{\left(\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(2 a \right)}}{4 \sin^{4}{\left(a \right)}}}{1 + \frac{\sin^{2}{\left(2 a \right)}}{4 \sin^{4}{\left(a \right)}}} & \text{otherwise} \end{cases}\right) + 1\right) \left(\left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(4 a \right)}}{4 \sin^{4}{\left(2 a \right)}}}{1 + \frac{\sin^{2}{\left(4 a \right)}}{4 \sin^{4}{\left(2 a \right)}}} & \text{otherwise} \end{cases}\right) + 1\right)}$$
// 0 for 4*a mod pi = 0\ // 1 for a mod pi = 0\
|| | || |
||/ 0 for 4*a mod pi = 0 | ||/ 1 for a mod pi = 0 |
||| | ||| |
|<| 2*cot(2*a) |*|<| 2 |
||<------------- otherwise otherwise | ||<-1 + cot (a) otherwise |
||| 2 | |||------------ otherwise |
|||1 + cot (2*a) | ||| 2 |
\\\ / \\\1 + cot (a) /
----------------------------------------------------------------------------------------------------------------------------------------------
/ // 1 for 2*a mod pi = 0\\
/ // 1 for a mod pi = 0\\ | || ||
| || || | || // 0 for 2*a mod pi = 0\ ||
| ||/ 1 for a mod pi = 0 || | || || | ||
| ||| || | || || 2 | ||
|1 + |<| 2 ||*|1 + |
$$\frac{\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{2 \cot{\left(2 a \right)}}{\cot^{2}{\left(2 a \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)}{\left(\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + 1\right) \left(\left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\left(\cot^{2}{\left(2 a \right)} - 1\right) \left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(a \right)}}{\left(\cot^{2}{\left(a \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) & \text{otherwise} \end{cases}\right) + 1\right)}$$
// 1 for a mod pi = 0\
|| |
// 0 for 4*a mod pi = 0\ || 2 |
|| | || cos (a) |
|| 2*cos(2*a) | ||-1 + ------------ |
||---------------------------------- otherwise | || 2/ pi\ |
||/ 2 \ | || cos |a - --| |
|<| cos (2*a) | / pi\ |*|< \ 2 / |
|||1 + --------------|*cos|2*a - --| | ||----------------- otherwise |
||| 2/ pi\| \ 2 / | || 2 |
||| cos |2*a - --|| | || cos (a) |
||\ \ 2 // | || 1 + ------------ |
\\ / || 2/ pi\ |
|| cos |a - --| |
\\ \ 2 / /
------------------------------------------------------------------------------------------------
/ // 1 for a mod pi = 0\\ / // 1 for 2*a mod pi = 0\\
| || || | || ||
| || 2 || | || 2 ||
| || cos (a) || | || cos (2*a) ||
| ||-1 + ------------ || | ||-1 + -------------- ||
| || 2/ pi\ || | || 2/ pi\ ||
| || cos |a - --| || | || cos |2*a - --| ||
|1 + |< \ 2 / ||*|1 + |< \ 2 / ||
| ||----------------- otherwise || | ||------------------- otherwise ||
| || 2 || | || 2 ||
| || cos (a) || | || cos (2*a) ||
| || 1 + ------------ || | || 1 + -------------- ||
| || 2/ pi\ || | || 2/ pi\ ||
| || cos |a - --| || | || cos |2*a - --| ||
\ \\ \ 2 / // \ \\ \ 2 / //
$$\frac{\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{2 \cos{\left(2 a \right)}}{\left(\frac{\cos^{2}{\left(2 a \right)}}{\cos^{2}{\left(2 a - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(2 a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\frac{\cos^{2}{\left(a \right)}}{\cos^{2}{\left(a - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(a \right)}}{\cos^{2}{\left(a - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)}{\left(\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\frac{\cos^{2}{\left(a \right)}}{\cos^{2}{\left(a - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(a \right)}}{\cos^{2}{\left(a - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) + 1\right) \left(\left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{\frac{\cos^{2}{\left(2 a \right)}}{\cos^{2}{\left(2 a - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(2 a \right)}}{\cos^{2}{\left(2 a - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) + 1\right)}$$
// 1 for a mod pi = 0\
// 0 for 4*a mod pi = 0\ || |
|| | || 2/ pi\ |
|| / pi\ | || sec |a - --| |
|| 2*sec|2*a - --| | || \ 2 / |
|| \ 2 / | ||-1 + ------------ |
||----------------------------- otherwise | || 2 |
|
$$\frac{\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{2 \sec{\left(2 a - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(2 a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(2 a \right)}}\right) \sec{\left(2 a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(a \right)}}}{1 + \frac{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(a \right)}}} & \text{otherwise} \end{cases}\right)}{\left(\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(a \right)}}}{1 + \frac{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(a \right)}}} & \text{otherwise} \end{cases}\right) + 1\right) \left(\left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(2 a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(2 a \right)}}}{1 + \frac{\sec^{2}{\left(2 a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(2 a \right)}}} & \text{otherwise} \end{cases}\right) + 1\right)}$$
// 1 for a mod pi = 0\
|| |
// 0 for 4*a mod pi = 0\ || 2 |
|| | || csc (a) |
|| 2*csc(2*a) | ||-1 + ------------ |
||---------------------------------- otherwise | || 2/pi \ |
||/ 2 \ | || csc |-- - a| |
|<| csc (2*a) | /pi \ |*|< \2 / |
|||1 + --------------|*csc|-- - 2*a| | ||----------------- otherwise |
||| 2/pi \| \2 / | || 2 |
||| csc |-- - 2*a|| | || csc (a) |
||\ \2 // | || 1 + ------------ |
\\ / || 2/pi \ |
|| csc |-- - a| |
\\ \2 / /
------------------------------------------------------------------------------------------------
/ // 1 for a mod pi = 0\\ / // 1 for 2*a mod pi = 0\\
| || || | || ||
| || 2 || | || 2 ||
| || csc (a) || | || csc (2*a) ||
| ||-1 + ------------ || | ||-1 + -------------- ||
| || 2/pi \ || | || 2/pi \ ||
| || csc |-- - a| || | || csc |-- - 2*a| ||
|1 + |< \2 / ||*|1 + |< \2 / ||
| ||----------------- otherwise || | ||------------------- otherwise ||
| || 2 || | || 2 ||
| || csc (a) || | || csc (2*a) ||
| || 1 + ------------ || | || 1 + -------------- ||
| || 2/pi \ || | || 2/pi \ ||
| || csc |-- - a| || | || csc |-- - 2*a| ||
\ \\ \2 / // \ \\ \2 / //
$$\frac{\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{2 \csc{\left(2 a \right)}}{\left(\frac{\csc^{2}{\left(2 a \right)}}{\csc^{2}{\left(- 2 a + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- 2 a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\frac{\csc^{2}{\left(a \right)}}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(a \right)}}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)}{\left(\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\frac{\csc^{2}{\left(a \right)}}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(a \right)}}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) + 1\right) \left(\left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{\frac{\csc^{2}{\left(2 a \right)}}{\csc^{2}{\left(- 2 a + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(2 a \right)}}{\csc^{2}{\left(- 2 a + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) + 1\right)}$$
Piecewise((0, Mod(4*a = pi, 0)), (2*csc(2*a)/((1 + csc(2*a)^2/csc(pi/2 - 2*a)^2)*csc(pi/2 - 2*a)), True))*Piecewise((1, Mod(a = pi, 0)), ((-1 + csc(a)^2/csc(pi/2 - a)^2)/(1 + csc(a)^2/csc(pi/2 - a)^2), True))/((1 + Piecewise((1, Mod(a = pi, 0)), ((-1 + csc(a)^2/csc(pi/2 - a)^2)/(1 + csc(a)^2/csc(pi/2 - a)^2), True)))*(1 + Piecewise((1, Mod(2*a = pi, 0)), ((-1 + csc(2*a)^2/csc(pi/2 - 2*a)^2)/(1 + csc(2*a)^2/csc(pi/2 - 2*a)^2), True))))