Господин Экзамен
cos(6*a)*cos(4*a)-sin(6*a)*sin(4*a) если a=4
Решение
Вы ввели
[src]
cos(6*a)*cos(4*a) - sin(6*a)*sin(4*a)
$$- \sin{\left(4 a \right)} \sin{\left(6 a \right)} + \cos{\left(4 a \right)} \cos{\left(6 a \right)}$$
cos(6*a)*cos(4*a) - sin(6*a)*sin(4*a)
Подстановка условия
[src]
cos(6*a)*cos(4*a) - sin(6*a)*sin(4*a) при a = 4
подставляем
cos(6*a)*cos(4*a) - sin(6*a)*sin(4*a)
$$- \sin{\left(4 a \right)} \sin{\left(6 a \right)} + \cos{\left(4 a \right)} \cos{\left(6 a \right)}$$
cos(4*a)*cos(6*a) - sin(4*a)*sin(6*a)
$$- \sin{\left(4 a \right)} \sin{\left(6 a \right)} + \cos{\left(4 a \right)} \cos{\left(6 a \right)}$$
переменные
a = 4
$$a = 4$$
cos(4*(4))*cos(6*(4)) - sin(4*(4))*sin(6*(4))
$$- \sin{\left(4 (4) \right)} \sin{\left(6 (4) \right)} + \cos{\left(4 (4) \right)} \cos{\left(6 (4) \right)}$$
cos(4*4)*cos(6*4) - sin(4*4)*sin(6*4)
$$\cos{\left(4 \cdot 4 \right)} \cos{\left(6 \cdot 4 \right)} - \sin{\left(4 \cdot 4 \right)} \sin{\left(6 \cdot 4 \right)}$$
cos(16)*cos(24) - sin(16)*sin(24)
$$\cos{\left(16 \right)} \cos{\left(24 \right)} - \sin{\left(16 \right)} \sin{\left(24 \right)}$$
cos(16)*cos(24) - sin(16)*sin(24)
Степени
[src]
/ -6*I*a 6*I*a\ / -4*I*a 4*I*a\ / -6*I*a 6*I*a\ / -4*I*a 4*I*a\ |e e | |e e | \- e + e /*\- e + e / |------- + ------|*|------- + ------| + ----------------------------------------- \ 2 2 / \ 2 2 / 4
$$\left(\frac{e^{4 i a}}{2} + \frac{e^{- 4 i a}}{2}\right) \left(\frac{e^{6 i a}}{2} + \frac{e^{- 6 i a}}{2}\right) + \frac{\left(e^{4 i a} - e^{- 4 i a}\right) \left(e^{6 i a} - e^{- 6 i a}\right)}{4}$$
(exp(-6*i*a)/2 + exp(6*i*a)/2)*(exp(-4*i*a)/2 + exp(4*i*a)/2) + (-exp(-6*i*a) + exp(6*i*a))*(-exp(-4*i*a) + exp(4*i*a))/4
Тригонометрическая часть
[src]
cos(10*a)
$$\cos{\left(10 a \right)}$$
1 --------- sec(10*a)
$$\frac{1}{\sec{\left(10 a \right)}}$$
/pi \ sin|-- + 10*a| \2 /
$$\sin{\left(10 a + \frac{\pi}{2} \right)}$$
1 -------------- /pi \ csc|-- - 10*a| \2 /
$$\frac{1}{\csc{\left(- 10 a + \frac{\pi}{2} \right)}}$$
2
1 - tan (5*a)
-------------
2
1 + tan (5*a)
$$\frac{- \tan^{2}{\left(5 a \right)} + 1}{\tan^{2}{\left(5 a \right)} + 1}$$
/ 1 for 5*a mod pi = 0 < \cos(10*a) otherwise
$$\begin{cases} 1 & \text{for}\: 5 a \bmod \pi = 0 \\\cos{\left(10 a \right)} & \text{otherwise} \end{cases}$$
cos(6*a) -------- - sin(4*a)*sin(6*a) sec(4*a)
$$- \sin{\left(4 a \right)} \sin{\left(6 a \right)} + \frac{\cos{\left(6 a \right)}}{\sec{\left(4 a \right)}}$$
/pi \
cos(6*a)*sin|-- + 4*a| - sin(4*a)*sin(6*a)
\2 /
$$- \sin{\left(4 a \right)} \sin{\left(6 a \right)} + \sin{\left(4 a + \frac{\pi}{2} \right)} \cos{\left(6 a \right)}$$
/pi \
cos(4*a)*sin|-- + 6*a| - sin(4*a)*sin(6*a)
\2 /
$$- \sin{\left(4 a \right)} \sin{\left(6 a \right)} + \sin{\left(6 a + \frac{\pi}{2} \right)} \cos{\left(4 a \right)}$$
1 1 ----------------- - ----------------- sec(4*a)*sec(6*a) csc(4*a)*csc(6*a)
$$\frac{1}{\sec{\left(4 a \right)} \sec{\left(6 a \right)}} - \frac{1}{\csc{\left(4 a \right)} \csc{\left(6 a \right)}}$$
cos(6*a) ------------- - sin(4*a)*sin(6*a) /pi \ csc|-- - 4*a| \2 /
$$- \sin{\left(4 a \right)} \sin{\left(6 a \right)} + \frac{\cos{\left(6 a \right)}}{\csc{\left(- 4 a + \frac{\pi}{2} \right)}}$$
/ pi\ / pi\
cos(4*a)*cos(6*a) - cos|4*a - --|*cos|6*a - --|
\ 2 / \ 2 /
$$\cos{\left(4 a \right)} \cos{\left(6 a \right)} - \cos{\left(4 a - \frac{\pi}{2} \right)} \cos{\left(6 a - \frac{\pi}{2} \right)}$$
/pi \ /pi \ sin|-- + 4*a|*sin|-- + 6*a| - sin(4*a)*sin(6*a) \2 / \2 /
$$- \sin{\left(4 a \right)} \sin{\left(6 a \right)} + \sin{\left(4 a + \frac{\pi}{2} \right)} \sin{\left(6 a + \frac{\pi}{2} \right)}$$
1 1 ---------------------- - ----------------- /pi \ csc(4*a)*csc(6*a) csc|-- - 6*a|*sec(4*a) \2 /
$$\frac{1}{\csc{\left(- 6 a + \frac{\pi}{2} \right)} \sec{\left(4 a \right)}} - \frac{1}{\csc{\left(4 a \right)} \csc{\left(6 a \right)}}$$
1 1
----------------- - ---------------------------
sec(4*a)*sec(6*a) / pi\ / pi\
sec|4*a - --|*sec|6*a - --|
\ 2 / \ 2 /
$$- \frac{1}{\sec{\left(4 a - \frac{\pi}{2} \right)} \sec{\left(6 a - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(4 a \right)} \sec{\left(6 a \right)}}$$
1 1 --------------------------- - ----------------- /pi \ /pi \ csc(4*a)*csc(6*a) csc|-- - 6*a|*csc|-- - 4*a| \2 / \2 /
$$\frac{1}{\csc{\left(- 6 a + \frac{\pi}{2} \right)} \csc{\left(- 4 a + \frac{\pi}{2} \right)}} - \frac{1}{\csc{\left(4 a \right)} \csc{\left(6 a \right)}}$$
/ 1 for 5*a mod pi = 0 | | 2 <-1 + cot (5*a) |-------------- otherwise | 2 \1 + cot (5*a)
$$\begin{cases} 1 & \text{for}\: 5 a \bmod \pi = 0 \\\frac{\cot^{2}{\left(5 a \right)} - 1}{\cot^{2}{\left(5 a \right)} + 1} & \text{otherwise} \end{cases}$$
1 1
----------------- - ---------------------------
sec(4*a)*sec(6*a) /pi \ /pi \
sec|-- - 6*a|*sec|-- - 4*a|
\2 / \2 /
$$- \frac{1}{\sec{\left(- 6 a + \frac{\pi}{2} \right)} \sec{\left(- 4 a + \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(4 a \right)} \sec{\left(6 a \right)}}$$
1 1 --------------------------- - --------------------------- /pi \ /pi \ csc(pi - 6*a)*csc(pi - 4*a) csc|-- - 6*a|*csc|-- - 4*a| \2 / \2 /
$$\frac{1}{\csc{\left(- 6 a + \frac{\pi}{2} \right)} \csc{\left(- 4 a + \frac{\pi}{2} \right)}} - \frac{1}{\csc{\left(- 6 a + \pi \right)} \csc{\left(- 4 a + \pi \right)}}$$
2 2 -cos(10*a) + cos(2*a) 2 2
1 - 2*cos (2*a) - 2*cos (3*a) - --------------------- + 4*cos (2*a)*cos (3*a)
2
$$4 \cos^{2}{\left(2 a \right)} \cos^{2}{\left(3 a \right)} - 2 \cos^{2}{\left(2 a \right)} - 2 \cos^{2}{\left(3 a \right)} - \frac{\cos{\left(2 a \right)} - \cos{\left(10 a \right)}}{2} + 1$$
/ 4 \
2 / 2 \ | 4*sin (2*a)|
cos (3*a)*\1 - tan (3*a)/*|1 - -----------|*(1 + cos(4*a))
| 2 |
-cos(10*a) + cos(2*a) \ sin (4*a) /
- --------------------- + ----------------------------------------------------------
2 2
$$\frac{\left(- \tan^{2}{\left(3 a \right)} + 1\right) \left(- \frac{4 \sin^{4}{\left(2 a \right)}}{\sin^{2}{\left(4 a \right)}} + 1\right) \left(\cos{\left(4 a \right)} + 1\right) \cos^{2}{\left(3 a \right)}}{2} - \frac{\cos{\left(2 a \right)} - \cos{\left(10 a \right)}}{2}$$
2/ pi\ / 2/ pi\\ / 2/ pi\\
sin |2*a + --|*|1 - cot |2*a + --||*|1 - cot |3*a + --||*(1 + sin(6*a))
cos(2*a) + cos(10*a) \ 4 / \ \ 4 // \ \ 4 //
-------------------- - -----------------------------------------------------------------------
2 2
$$- \frac{\left(- \cot^{2}{\left(2 a + \frac{\pi}{4} \right)} + 1\right) \left(- \cot^{2}{\left(3 a + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(6 a \right)} + 1\right) \sin^{2}{\left(2 a + \frac{\pi}{4} \right)}}{2} + \frac{\cos{\left(2 a \right)} + \cos{\left(10 a \right)}}{2}$$
/ 2 \ / 2 \ \1 - tan (2*a)/*\1 - tan (3*a)/ 4*tan(2*a)*tan(3*a) ------------------------------- - ------------------------------- / 2 \ / 2 \ / 2 \ / 2 \ \1 + tan (2*a)/*\1 + tan (3*a)/ \1 + tan (2*a)/*\1 + tan (3*a)/
$$\frac{\left(- \tan^{2}{\left(2 a \right)} + 1\right) \left(- \tan^{2}{\left(3 a \right)} + 1\right)}{\left(\tan^{2}{\left(2 a \right)} + 1\right) \left(\tan^{2}{\left(3 a \right)} + 1\right)} - \frac{4 \tan{\left(2 a \right)} \tan{\left(3 a \right)}}{\left(\tan^{2}{\left(2 a \right)} + 1\right) \left(\tan^{2}{\left(3 a \right)} + 1\right)}$$
/ 2 \ / pi\
2*\1 - tan (3*a)/*tan|2*a + --|
4*tan(2*a)*tan(3*a) \ 4 /
- ------------------------------- + ------------------------------------
/ 2 \ / 2 \ / 2 \ / 2/ pi\\
\1 + tan (2*a)/*\1 + tan (3*a)/ \1 + tan (3*a)/*|1 + tan |2*a + --||
\ \ 4 //
$$\frac{2 \cdot \left(- \tan^{2}{\left(3 a \right)} + 1\right) \tan{\left(2 a + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(3 a \right)} + 1\right) \left(\tan^{2}{\left(2 a + \frac{\pi}{4} \right)} + 1\right)} - \frac{4 \tan{\left(2 a \right)} \tan{\left(3 a \right)}}{\left(\tan^{2}{\left(2 a \right)} + 1\right) \left(\tan^{2}{\left(3 a \right)} + 1\right)}$$
/ pi\ / pi\
4*tan|2*a + --|*tan|3*a + --|
4*tan(2*a)*tan(3*a) \ 4 / \ 4 /
- ------------------------------- + -----------------------------------------
/ 2 \ / 2 \ / 2/ pi\\ / 2/ pi\\
\1 + tan (2*a)/*\1 + tan (3*a)/ |1 + tan |2*a + --||*|1 + tan |3*a + --||
\ \ 4 // \ \ 4 //
$$\frac{4 \tan{\left(2 a + \frac{\pi}{4} \right)} \tan{\left(3 a + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(2 a + \frac{\pi}{4} \right)} + 1\right) \left(\tan^{2}{\left(3 a + \frac{\pi}{4} \right)} + 1\right)} - \frac{4 \tan{\left(2 a \right)} \tan{\left(3 a \right)}}{\left(\tan^{2}{\left(2 a \right)} + 1\right) \left(\tan^{2}{\left(3 a \right)} + 1\right)}$$
/ pi\ / pi\
4*tan|2*a + --|*tan|3*a + --|
4*cot(2*a)*cot(3*a) \ 4 / \ 4 /
- ------------------------------- + -----------------------------------------
/ 2 \ / 2 \ / 2/ pi\\ / 2/ pi\\
\1 + cot (2*a)/*\1 + cot (3*a)/ |1 + tan |2*a + --||*|1 + tan |3*a + --||
\ \ 4 // \ \ 4 //
$$- \frac{4 \cot{\left(2 a \right)} \cot{\left(3 a \right)}}{\left(\cot^{2}{\left(2 a \right)} + 1\right) \left(\cot^{2}{\left(3 a \right)} + 1\right)} + \frac{4 \tan{\left(2 a + \frac{\pi}{4} \right)} \tan{\left(3 a + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(2 a + \frac{\pi}{4} \right)} + 1\right) \left(\tan^{2}{\left(3 a + \frac{\pi}{4} \right)} + 1\right)}$$
/ 2 \ |/ 2 2 \ 2 2 | / 2 2 \ / 2 2 \ \\cos (a) - sin (a)/ - 4*cos (a)*sin (a)/*\cos (3*a) - sin (3*a)/ - 8*\cos (a) - sin (a)/*cos(a)*cos(3*a)*sin(a)*sin(3*a)
$$- 8 \left(- \sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}\right) \sin{\left(a \right)} \sin{\left(3 a \right)} \cos{\left(a \right)} \cos{\left(3 a \right)} + \left(- 4 \sin^{2}{\left(a \right)} \cos^{2}{\left(a \right)} + \left(- \sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}\right)^{2}\right) \left(- \sin^{2}{\left(3 a \right)} + \cos^{2}{\left(3 a \right)}\right)$$
/ 1 \ / 1 \ |1 - ---------|*|1 - ---------| | 2 | | 2 | \ cot (2*a)/ \ cot (3*a)/ 4 ------------------------------- - ------------------------------------------------- / 1 \ / 1 \ / 1 \ / 1 \ |1 + ---------|*|1 + ---------| |1 + ---------|*|1 + ---------|*cot(2*a)*cot(3*a) | 2 | | 2 | | 2 | | 2 | \ cot (2*a)/ \ cot (3*a)/ \ cot (2*a)/ \ cot (3*a)/
$$\frac{\left(1 - \frac{1}{\cot^{2}{\left(2 a \right)}}\right) \left(1 - \frac{1}{\cot^{2}{\left(3 a \right)}}\right)}{\left(1 + \frac{1}{\cot^{2}{\left(2 a \right)}}\right) \left(1 + \frac{1}{\cot^{2}{\left(3 a \right)}}\right)} - \frac{4}{\left(1 + \frac{1}{\cot^{2}{\left(2 a \right)}}\right) \left(1 + \frac{1}{\cot^{2}{\left(3 a \right)}}\right) \cot{\left(2 a \right)} \cot{\left(3 a \right)}}$$
/ 2/ pi\\ / 2/ pi\\
/ 2 \ / 2 \ |-1 + tan |2*a + --||*|-1 + tan |3*a + --||
\-1 + cot (2*a)/*\-1 + cot (3*a)/ \ \ 4 // \ \ 4 //
--------------------------------- - -------------------------------------------
/ 2 \ / 2 \ / 2/ pi\\ / 2/ pi\\
\1 + cot (2*a)/*\1 + cot (3*a)/ |1 + tan |2*a + --||*|1 + tan |3*a + --||
\ \ 4 // \ \ 4 //
$$- \frac{\left(\tan^{2}{\left(2 a + \frac{\pi}{4} \right)} - 1\right) \left(\tan^{2}{\left(3 a + \frac{\pi}{4} \right)} - 1\right)}{\left(\tan^{2}{\left(2 a + \frac{\pi}{4} \right)} + 1\right) \left(\tan^{2}{\left(3 a + \frac{\pi}{4} \right)} + 1\right)} + \frac{\left(\cot^{2}{\left(2 a \right)} - 1\right) \left(\cot^{2}{\left(3 a \right)} - 1\right)}{\left(\cot^{2}{\left(2 a \right)} + 1\right) \left(\cot^{2}{\left(3 a \right)} + 1\right)}$$
/ 2/ pi\\ / 2/ pi\\
/ 2 \ / 2 \ |1 - cot |2*a + --||*|1 - cot |3*a + --||
\1 - tan (2*a)/*\1 - tan (3*a)/ \ \ 4 // \ \ 4 //
------------------------------- - -----------------------------------------
/ 2 \ / 2 \ / 2/ pi\\ / 2/ pi\\
\1 + tan (2*a)/*\1 + tan (3*a)/ |1 + cot |2*a + --||*|1 + cot |3*a + --||
\ \ 4 // \ \ 4 //
$$\frac{\left(- \tan^{2}{\left(2 a \right)} + 1\right) \left(- \tan^{2}{\left(3 a \right)} + 1\right)}{\left(\tan^{2}{\left(2 a \right)} + 1\right) \left(\tan^{2}{\left(3 a \right)} + 1\right)} - \frac{\left(- \cot^{2}{\left(2 a + \frac{\pi}{4} \right)} + 1\right) \left(- \cot^{2}{\left(3 a + \frac{\pi}{4} \right)} + 1\right)}{\left(\cot^{2}{\left(2 a + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(3 a + \frac{\pi}{4} \right)} + 1\right)}$$
// 0 for 4*a mod pi = 0\ // 0 for 6*a mod pi = 0\ 2 / 2 \ // 1 for 3*a mod pi = 0\ - |< |*|< | + sin (2*a)*\-1 + cot (2*a)/*|< | \\sin(4*a) otherwise / \\sin(6*a) otherwise / \\cos(6*a) otherwise /
$$\left(\left(\cot^{2}{\left(2 a \right)} - 1\right) \left(\begin{cases} 1 & \text{for}\: 3 a \bmod \pi = 0 \\\cos{\left(6 a \right)} & \text{otherwise} \end{cases}\right) \sin^{2}{\left(2 a \right)}\right) - \left(\left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\sin{\left(6 a \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 3*a mod pi = 0\ / pi\
2*|< |*tan|2*a + --|
// 0 for 4*a mod pi = 0\ // 0 for 6*a mod pi = 0\ \\cos(6*a) otherwise / \ 4 /
- |< |*|< | + -----------------------------------------------
\\sin(4*a) otherwise / \\sin(6*a) otherwise / 2/ pi\
1 + tan |2*a + --|
\ 4 /
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\sin{\left(6 a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\frac{2 \left(\begin{cases} 1 & \text{for}\: 3 a \bmod \pi = 0 \\\cos{\left(6 a \right)} & \text{otherwise} \end{cases}\right) \tan{\left(2 a + \frac{\pi}{4} \right)}}{\tan^{2}{\left(2 a + \frac{\pi}{4} \right)} + 1}\right)$$
// /pi \ \ || 0 for |-- + 4*a| mod pi = 0| // 1 for 3*a mod pi = 0\ // 0 for 4*a mod pi = 0\ // 0 for 6*a mod pi = 0\ |< \2 / |*|< | - |< |*|< | || | \\cos(6*a) otherwise / \\sin(4*a) otherwise / \\sin(6*a) otherwise / \\cos(4*a) otherwise /
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\sin{\left(6 a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 0 & \text{for}\: \left(4 a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\cos{\left(4 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 3 a \bmod \pi = 0 \\\cos{\left(6 a \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 2*a mod pi = 0\ || | // 1 for 3*a mod pi = 0\ // 0 for 4*a mod pi = 0\ // 0 for 6*a mod pi = 0\ |< 2 / 2 \ |*|< | - |< |*|< | ||sin (2*a)*\-1 + cot (2*a)/ otherwise | \\cos(6*a) otherwise / \\sin(4*a) otherwise / \\sin(6*a) otherwise / \\ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\sin{\left(6 a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\left(\cot^{2}{\left(2 a \right)} - 1\right) \sin^{2}{\left(2 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 3 a \bmod \pi = 0 \\\cos{\left(6 a \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 2*a mod pi = 0\ || | || 1 | ||-1 + --------- | // 1 for 3*a mod pi = 0\ // 0 for 4*a mod pi = 0\ // 0 for 6*a mod pi = 0\ |< 2 |*|< | - |< |*|< | || tan (2*a) | \\cos(6*a) otherwise / \\sin(4*a) otherwise / \\sin(6*a) otherwise / ||-------------- otherwise | || 2 | \\ csc (2*a) /
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\sin{\left(6 a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(2 a \right)}}}{\csc^{2}{\left(2 a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 3 a \bmod \pi = 0 \\\cos{\left(6 a \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 2*a mod pi = 0\ || | || 2 / 1 \ | // 1 for 3*a mod pi = 0\ // 0 for 4*a mod pi = 0\ // 0 for 6*a mod pi = 0\ |
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\sin{\left(6 a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\left(-1 + \frac{1}{\tan^{2}{\left(2 a \right)}}\right) \sin^{2}{\left(2 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 3 a \bmod \pi = 0 \\\cos{\left(6 a \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 2*a mod pi = 0\ || | || 4 2 / 1 \ | // 1 for 3*a mod pi = 0\ // 0 for 4*a mod pi = 0\ // 0 for 6*a mod pi = 0\ |<4*cos (a)*tan (a)*|-1 + ---------| otherwise |*|< | - |< |*|< | || | 2 | | \\cos(6*a) otherwise / \\sin(4*a) otherwise / \\sin(6*a) otherwise / || \ tan (2*a)/ | \\ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\sin{\left(6 a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\4 \left(-1 + \frac{1}{\tan^{2}{\left(2 a \right)}}\right) \cos^{4}{\left(a \right)} \tan^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 3 a \bmod \pi = 0 \\\cos{\left(6 a \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 2*a mod pi = 0\ || | || / 2 \ | // 1 for 3*a mod pi = 0\ // 0 for 4*a mod pi = 0\ // 0 for 6*a mod pi = 0\ |< 2 | sin (4*a) | |*|< | - |< |*|< | ||sin (2*a)*|-1 + -----------| otherwise | \\cos(6*a) otherwise / \\sin(4*a) otherwise / \\sin(6*a) otherwise / || | 4 | | \\ \ 4*sin (2*a)/ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\sin{\left(6 a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\left(-1 + \frac{\sin^{2}{\left(4 a \right)}}{4 \sin^{4}{\left(2 a \right)}}\right) \sin^{2}{\left(2 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 3 a \bmod \pi = 0 \\\cos{\left(6 a \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 3*a mod pi = 0\
|| |
/ 2 \ || 2 |
\-1 + cot (2*a)/*|<-1 + cot (3*a) |
// 0 for 4*a mod pi = 0\ // 0 for 6*a mod pi = 0\ ||-------------- otherwise |
|| | || | || 2 |
|| 2*cot(2*a) | || 2*cot(3*a) | \\1 + cot (3*a) /
- |<------------- otherwise |*|<------------- otherwise | + ------------------------------------------------------
|| 2 | || 2 | 2
||1 + cot (2*a) | ||1 + cot (3*a) | 1 + cot (2*a)
\\ / \\ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{2 \cot{\left(2 a \right)}}{\cot^{2}{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\frac{2 \cot{\left(3 a \right)}}{\cot^{2}{\left(3 a \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\frac{\left(\cot^{2}{\left(2 a \right)} - 1\right) \left(\begin{cases} 1 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{\cot^{2}{\left(3 a \right)} - 1}{\cot^{2}{\left(3 a \right)} + 1} & \text{otherwise} \end{cases}\right)}{\cot^{2}{\left(2 a \right)} + 1}\right)$$
// 1 for 2*a mod pi = 0\ || | // 1 for 3*a mod pi = 0\ || / 2 \ | || | // 0 for 4*a mod pi = 0\ // 0 for 6*a mod pi = 0\ |< 2 | sin (4*a) | |*|< /pi \ | - |< |*|< | ||sin (2*a)*|-1 + -----------| otherwise | ||sin|-- + 6*a| otherwise | \\sin(4*a) otherwise / \\sin(6*a) otherwise / || | 4 | | \\ \2 / / \\ \ 4*sin (2*a)/ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\sin{\left(6 a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\left(-1 + \frac{\sin^{2}{\left(4 a \right)}}{4 \sin^{4}{\left(2 a \right)}}\right) \sin^{2}{\left(2 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 3 a \bmod \pi = 0 \\\sin{\left(6 a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 3*a mod pi = 0\
|| |
|| 2 | / pi\
2*|<-1 + cot (3*a) |*tan|2*a + --|
// 0 for 4*a mod pi = 0\ // 0 for 6*a mod pi = 0\ ||-------------- otherwise | \ 4 /
|| | || | || 2 |
|| 2*cot(2*a) | || 2*cot(3*a) | \\1 + cot (3*a) /
- |<------------- otherwise |*|<------------- otherwise | + -----------------------------------------------------
|| 2 | || 2 | 2/ pi\
||1 + cot (2*a) | ||1 + cot (3*a) | 1 + tan |2*a + --|
\\ / \\ / \ 4 /
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{2 \cot{\left(2 a \right)}}{\cot^{2}{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\frac{2 \cot{\left(3 a \right)}}{\cot^{2}{\left(3 a \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\frac{2 \left(\begin{cases} 1 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{\cot^{2}{\left(3 a \right)} - 1}{\cot^{2}{\left(3 a \right)} + 1} & \text{otherwise} \end{cases}\right) \tan{\left(2 a + \frac{\pi}{4} \right)}}{\tan^{2}{\left(2 a + \frac{\pi}{4} \right)} + 1}\right)$$
// 1 for 2*a mod pi = 0\ // / 3*pi\ \ // / 3*pi\ \ || | // 1 for 3*a mod pi = 0\ || 1 for |4*a + ----| mod 2*pi = 0| || 1 for |6*a + ----| mod 2*pi = 0| |< 2 / 2 \ |*|< | - |< \ 2 / |*|< \ 2 / | ||sin (2*a)*\-1 + cot (2*a)/ otherwise | \\cos(6*a) otherwise / || | || | \\ / \\sin(4*a) otherwise / \\sin(6*a) otherwise /
$$\left(\left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\left(\cot^{2}{\left(2 a \right)} - 1\right) \sin^{2}{\left(2 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 3 a \bmod \pi = 0 \\\cos{\left(6 a \right)} & \text{otherwise} \end{cases}\right)\right) - \left(\left(\begin{cases} 1 & \text{for}\: \left(4 a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(6 a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(6 a \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 2*a mod pi = 0\ || | || / 2 \ | || | sin (4*a) | | // 1 for 3*a mod pi = 0\ // 0 for 4*a mod pi = 0\ // 0 for 6*a mod pi = 0\ |<(1 - cos(4*a))*|-1 + -----------| |*|< | - |< |*|< | || | 4 | | \\cos(6*a) otherwise / \\sin(4*a) otherwise / \\sin(6*a) otherwise / || \ 4*sin (2*a)/ | ||--------------------------------- otherwise | \\ 2 /
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\sin{\left(6 a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{\left(-1 + \frac{\sin^{2}{\left(4 a \right)}}{4 \sin^{4}{\left(2 a \right)}}\right) \left(- \cos{\left(4 a \right)} + 1\right)}{2} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 3 a \bmod \pi = 0 \\\cos{\left(6 a \right)} & \text{otherwise} \end{cases}\right)\right)$$
// /pi \ \
// /pi \ \ || 0 for |-- + 6*a| mod pi = 0|
|| 0 for |-- + 4*a| mod pi = 0| || \2 / | // 0 for 4*a mod pi = 0\ // 0 for 6*a mod pi = 0\
|< \2 / |*|< | - |< |*|< |
|| | || / pi\ | \\sin(4*a) otherwise / \\sin(6*a) otherwise /
\\cos(4*a) otherwise / ||(1 + sin(6*a))*cot|3*a + --| otherwise |
\\ \ 4 / /
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\sin{\left(6 a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 0 & \text{for}\: \left(4 a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\cos{\left(4 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(6 a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(6 a \right)} + 1\right) \cot{\left(3 a + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)\right)$$
/ 4 \ / 4 \ | 4*sin (2*a)| | 4*sin (3*a)| |1 - -----------|*|1 - -----------| | 2 | | 2 | 2 2 \ sin (4*a) / \ sin (6*a) / 16*sin (2*a)*sin (3*a) ----------------------------------- - ----------------------------------------------------- / 4 \ / 4 \ / 4 \ / 4 \ | 4*sin (2*a)| | 4*sin (3*a)| | 4*sin (2*a)| | 4*sin (3*a)| |1 + -----------|*|1 + -----------| |1 + -----------|*|1 + -----------|*sin(4*a)*sin(6*a) | 2 | | 2 | | 2 | | 2 | \ sin (4*a) / \ sin (6*a) / \ sin (4*a) / \ sin (6*a) /
$$\frac{\left(- \frac{4 \sin^{4}{\left(2 a \right)}}{\sin^{2}{\left(4 a \right)}} + 1\right) \left(- \frac{4 \sin^{4}{\left(3 a \right)}}{\sin^{2}{\left(6 a \right)}} + 1\right)}{\left(\frac{4 \sin^{4}{\left(2 a \right)}}{\sin^{2}{\left(4 a \right)}} + 1\right) \left(\frac{4 \sin^{4}{\left(3 a \right)}}{\sin^{2}{\left(6 a \right)}} + 1\right)} - \frac{16 \sin^{2}{\left(2 a \right)} \sin^{2}{\left(3 a \right)}}{\left(\frac{4 \sin^{4}{\left(2 a \right)}}{\sin^{2}{\left(4 a \right)}} + 1\right) \left(\frac{4 \sin^{4}{\left(3 a \right)}}{\sin^{2}{\left(6 a \right)}} + 1\right) \sin{\left(4 a \right)} \sin{\left(6 a \right)}}$$
// 1 for 2*a mod pi = 0\ || | || 2 | // 1 for 3*a mod pi = 0\ || csc (2*a) | || | // 0 for 4*a mod pi = 0\ // 0 for 6*a mod pi = 0\ ||-1 + -------------- | || 1 | || | || | |< 2/pi \ |*|<------------- otherwise | - |< 1 |*|< 1 | || csc |-- - 2*a| | || /pi \ | ||-------- otherwise | ||-------- otherwise | || \2 / | ||csc|-- - 6*a| | \\csc(4*a) / \\csc(6*a) / ||------------------- otherwise | \\ \2 / / || 2 | \\ csc (2*a) /
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{1}{\csc{\left(4 a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\frac{1}{\csc{\left(6 a \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{\frac{\csc^{2}{\left(2 a \right)}}{\csc^{2}{\left(- 2 a + \frac{\pi}{2} \right)}} - 1}{\csc^{2}{\left(2 a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{1}{\csc{\left(- 6 a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 2*a mod pi = 0\ || | || / 2 \ | // 0 for 4*a mod pi = 0\ // 0 for 6*a mod pi = 0\ || 2/ pi\ | cos (2*a) | | // 1 for 3*a mod pi = 0\ || | || | |
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\cos{\left(4 a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\cos{\left(6 a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\left(\frac{\cos^{2}{\left(2 a \right)}}{\cos^{2}{\left(2 a - \frac{\pi}{2} \right)}} - 1\right) \cos^{2}{\left(2 a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 3 a \bmod \pi = 0 \\\cos{\left(6 a \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 2*a mod pi = 0\ || | || 2/ pi\ | || sec |2*a - --| | // 0 for 4*a mod pi = 0\ // 0 for 6*a mod pi = 0\ || \ 2 / | // 1 for 3*a mod pi = 0\ || | || | ||-1 + -------------- | || | || 1 | || 1 | |< 2 |*|< 1 | - |<------------- otherwise |*|<------------- otherwise | || sec (2*a) | ||-------- otherwise | || / pi\ | || / pi\ | ||------------------- otherwise | \\sec(6*a) / ||sec|4*a - --| | ||sec|6*a - --| | || 2/ pi\ | \\ \ 2 / / \\ \ 2 / / || sec |2*a - --| | || \ 2 / | \\ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{1}{\sec{\left(4 a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\frac{1}{\sec{\left(6 a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(2 a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(2 a \right)}}}{\sec^{2}{\left(2 a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{1}{\sec{\left(6 a \right)}} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 2*a mod pi = 0\ // 1 for 3*a mod pi = 0\ // 0 for 4*a mod pi = 0\ // 0 for 6*a mod pi = 0\ || | || | || | || | || 2 | || 2 | || 2*cot(2*a) | || 2*cot(3*a) | |<-1 + cot (2*a) |*|<-1 + cot (3*a) | - |<------------- otherwise |*|<------------- otherwise | ||-------------- otherwise | ||-------------- otherwise | || 2 | || 2 | || 2 | || 2 | ||1 + cot (2*a) | ||1 + cot (3*a) | \\1 + cot (2*a) / \\1 + cot (3*a) / \\ / \\ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{2 \cot{\left(2 a \right)}}{\cot^{2}{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\frac{2 \cot{\left(3 a \right)}}{\cot^{2}{\left(3 a \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{\cot^{2}{\left(2 a \right)} - 1}{\cot^{2}{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{\cot^{2}{\left(3 a \right)} - 1}{\cot^{2}{\left(3 a \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
/ 2/ pi\\ / 2/ pi\\ | cos |2*a - --|| | cos |3*a - --|| | \ 2 /| | \ 2 /| |1 - --------------|*|1 - --------------| / pi\ / pi\ | 2 | | 2 | 4*cos|2*a - --|*cos|3*a - --| \ cos (2*a) / \ cos (3*a) / \ 2 / \ 2 / ----------------------------------------- - ----------------------------------------------------------- / 2/ pi\\ / 2/ pi\\ / 2/ pi\\ / 2/ pi\\ | cos |2*a - --|| | cos |3*a - --|| | cos |2*a - --|| | cos |3*a - --|| | \ 2 /| | \ 2 /| | \ 2 /| | \ 2 /| |1 + --------------|*|1 + --------------| |1 + --------------|*|1 + --------------|*cos(2*a)*cos(3*a) | 2 | | 2 | | 2 | | 2 | \ cos (2*a) / \ cos (3*a) / \ cos (2*a) / \ cos (3*a) /
$$\frac{\left(1 - \frac{\cos^{2}{\left(2 a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(2 a \right)}}\right) \left(1 - \frac{\cos^{2}{\left(3 a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(3 a \right)}}\right)}{\left(1 + \frac{\cos^{2}{\left(2 a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(2 a \right)}}\right) \left(1 + \frac{\cos^{2}{\left(3 a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(3 a \right)}}\right)} - \frac{4 \cos{\left(2 a - \frac{\pi}{2} \right)} \cos{\left(3 a - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(2 a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(2 a \right)}}\right) \left(1 + \frac{\cos^{2}{\left(3 a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(3 a \right)}}\right) \cos{\left(2 a \right)} \cos{\left(3 a \right)}}$$
/ 2 \ / 2 \ | sec (2*a) | | sec (3*a) | |1 - --------------|*|1 - --------------| | 2/ pi\| | 2/ pi\| | sec |2*a - --|| | sec |3*a - --|| \ \ 2 // \ \ 2 // 4*sec(2*a)*sec(3*a) ----------------------------------------- - --------------------------------------------------------------------- / 2 \ / 2 \ / 2 \ / 2 \ | sec (2*a) | | sec (3*a) | | sec (2*a) | | sec (3*a) | / pi\ / pi\ |1 + --------------|*|1 + --------------| |1 + --------------|*|1 + --------------|*sec|2*a - --|*sec|3*a - --| | 2/ pi\| | 2/ pi\| | 2/ pi\| | 2/ pi\| \ 2 / \ 2 / | sec |2*a - --|| | sec |3*a - --|| | sec |2*a - --|| | sec |3*a - --|| \ \ 2 // \ \ 2 // \ \ 2 // \ \ 2 //
$$\frac{\left(- \frac{\sec^{2}{\left(2 a \right)}}{\sec^{2}{\left(2 a - \frac{\pi}{2} \right)}} + 1\right) \left(- \frac{\sec^{2}{\left(3 a \right)}}{\sec^{2}{\left(3 a - \frac{\pi}{2} \right)}} + 1\right)}{\left(\frac{\sec^{2}{\left(2 a \right)}}{\sec^{2}{\left(2 a - \frac{\pi}{2} \right)}} + 1\right) \left(\frac{\sec^{2}{\left(3 a \right)}}{\sec^{2}{\left(3 a - \frac{\pi}{2} \right)}} + 1\right)} - \frac{4 \sec{\left(2 a \right)} \sec{\left(3 a \right)}}{\left(\frac{\sec^{2}{\left(2 a \right)}}{\sec^{2}{\left(2 a - \frac{\pi}{2} \right)}} + 1\right) \left(\frac{\sec^{2}{\left(3 a \right)}}{\sec^{2}{\left(3 a - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(2 a - \frac{\pi}{2} \right)} \sec{\left(3 a - \frac{\pi}{2} \right)}}$$
// /pi \ \ || 0 for |-- + 4*a| mod pi = 0| || \2 / | // 1 for 3*a mod pi = 0\ // 0 for 4*a mod pi = 0\ // 0 for 6*a mod pi = 0\ || | || | || | || | || / pi\ | || 2 | || 2*cot(2*a) | || 2*cot(3*a) | |< 2*cot|2*a + --| |*|<-1 + cot (3*a) | - |<------------- otherwise |*|<------------- otherwise | || \ 4 / | ||-------------- otherwise | || 2 | || 2 | ||------------------ otherwise | || 2 | ||1 + cot (2*a) | ||1 + cot (3*a) | || 2/ pi\ | \\1 + cot (3*a) / \\ / \\ / ||1 + cot |2*a + --| | \\ \ 4 / /
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{2 \cot{\left(2 a \right)}}{\cot^{2}{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\frac{2 \cot{\left(3 a \right)}}{\cot^{2}{\left(3 a \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 0 & \text{for}\: \left(4 a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(2 a + \frac{\pi}{4} \right)}}{\cot^{2}{\left(2 a + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{\cot^{2}{\left(3 a \right)} - 1}{\cot^{2}{\left(3 a \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
/ 2/pi \\ / 2/pi \\ | csc |-- - 2*a|| | csc |-- - 3*a|| | \2 /| | \2 /| |1 - --------------|*|1 - --------------| /pi \ /pi \ | 2 | | 2 | 4*csc|-- - 3*a|*csc|-- - 2*a| \ csc (2*a) / \ csc (3*a) / \2 / \2 / ----------------------------------------- - ----------------------------------------------------------- / 2/pi \\ / 2/pi \\ / 2/pi \\ / 2/pi \\ | csc |-- - 2*a|| | csc |-- - 3*a|| | csc |-- - 2*a|| | csc |-- - 3*a|| | \2 /| | \2 /| | \2 /| | \2 /| |1 + --------------|*|1 + --------------| |1 + --------------|*|1 + --------------|*csc(2*a)*csc(3*a) | 2 | | 2 | | 2 | | 2 | \ csc (2*a) / \ csc (3*a) / \ csc (2*a) / \ csc (3*a) /
$$\frac{\left(1 - \frac{\csc^{2}{\left(- 2 a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(2 a \right)}}\right) \left(1 - \frac{\csc^{2}{\left(- 3 a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(3 a \right)}}\right)}{\left(1 + \frac{\csc^{2}{\left(- 2 a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(2 a \right)}}\right) \left(1 + \frac{\csc^{2}{\left(- 3 a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(3 a \right)}}\right)} - \frac{4 \csc{\left(- 3 a + \frac{\pi}{2} \right)} \csc{\left(- 2 a + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- 2 a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(2 a \right)}}\right) \left(1 + \frac{\csc^{2}{\left(- 3 a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(3 a \right)}}\right) \csc{\left(2 a \right)} \csc{\left(3 a \right)}}$$
// 1 for 2*a mod pi = 0\ || | || 2 / 1 \ | // 1 for 3*a mod pi = 0\ // 0 for 4*a mod pi = 0\ // 0 for 6*a mod pi = 0\ ||4*tan (a)*|-1 + ---------| | || | || | || | || | 2 | | || 2 | || 2*tan(2*a) | || 2*tan(3*a) | |< \ tan (2*a)/ |*|<1 - tan (3*a) | - |<------------- otherwise |*|<------------- otherwise | ||-------------------------- otherwise | ||------------- otherwise | || 2 | || 2 | || 2 | || 2 | ||1 + tan (2*a) | ||1 + tan (3*a) | || / 2 \ | \\1 + tan (3*a) / \\ / \\ / || \1 + tan (a)/ | \\ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{2 \tan{\left(2 a \right)}}{\tan^{2}{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\frac{2 \tan{\left(3 a \right)}}{\tan^{2}{\left(3 a \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{4 \left(-1 + \frac{1}{\tan^{2}{\left(2 a \right)}}\right) \tan^{2}{\left(a \right)}}{\left(\tan^{2}{\left(a \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{- \tan^{2}{\left(3 a \right)} + 1}{\tan^{2}{\left(3 a \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 2*a mod pi = 0\ // 1 for 3*a mod pi = 0\
|| | || | // 0 for 4*a mod pi = 0\ // 0 for 6*a mod pi = 0\
|| 1 | || 1 | || | || |
||-1 + --------- | ||-1 + --------- | || 2 | || 2 |
|| 2 | || 2 | ||------------------------ otherwise | ||------------------------ otherwise |
|< tan (2*a) |*|< tan (3*a) | - |
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(2 a \right)}}\right) \tan{\left(2 a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(3 a \right)}}\right) \tan{\left(3 a \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(2 a \right)}}}{1 + \frac{1}{\tan^{2}{\left(2 a \right)}}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(3 a \right)}}}{1 + \frac{1}{\tan^{2}{\left(3 a \right)}}} & \text{otherwise} \end{cases}\right)\right)$$
// /pi \ \ // /pi \ \ || 0 for |-- + 4*a| mod pi = 0| || 0 for |-- + 6*a| mod pi = 0| || \2 / | || \2 / | // 0 for 4*a mod pi = 0\ // 0 for 6*a mod pi = 0\ || | || | || | || | || / pi\ | || / pi\ | || 2*cot(2*a) | || 2*cot(3*a) | |< 2*cot|2*a + --| |*|< 2*cot|3*a + --| | - |<------------- otherwise |*|<------------- otherwise | || \ 4 / | || \ 4 / | || 2 | || 2 | ||------------------ otherwise | ||------------------ otherwise | ||1 + cot (2*a) | ||1 + cot (3*a) | || 2/ pi\ | || 2/ pi\ | \\ / \\ / ||1 + cot |2*a + --| | ||1 + cot |3*a + --| | \\ \ 4 / / \\ \ 4 / /
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{2 \cot{\left(2 a \right)}}{\cot^{2}{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\frac{2 \cot{\left(3 a \right)}}{\cot^{2}{\left(3 a \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 0 & \text{for}\: \left(4 a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(2 a + \frac{\pi}{4} \right)}}{\cot^{2}{\left(2 a + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(6 a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(3 a + \frac{\pi}{4} \right)}}{\cot^{2}{\left(3 a + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
// / 3*pi\ \ // / 3*pi\ \
|| 1 for |4*a + ----| mod 2*pi = 0| || 1 for |6*a + ----| mod 2*pi = 0|
// 1 for 2*a mod pi = 0\ // 1 for 3*a mod pi = 0\ || \ 2 / | || \ 2 / |
|| | || | || | || |
|| 2 | || 2 | || 2/ pi\ | || 2/ pi\ |
|<-1 + cot (2*a) |*|<-1 + cot (3*a) | - |<-1 + tan |2*a + --| |*|<-1 + tan |3*a + --| |
||-------------- otherwise | ||-------------- otherwise | || \ 4 / | || \ 4 / |
|| 2 | || 2 | ||------------------- otherwise | ||------------------- otherwise |
\\1 + cot (2*a) / \\1 + cot (3*a) / || 2/ pi\ | || 2/ pi\ |
|| 1 + tan |2*a + --| | || 1 + tan |3*a + --| |
\\ \ 4 / / \\ \ 4 / /
$$\left(\left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{\cot^{2}{\left(2 a \right)} - 1}{\cot^{2}{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{\cot^{2}{\left(3 a \right)} - 1}{\cot^{2}{\left(3 a \right)} + 1} & \text{otherwise} \end{cases}\right)\right) - \left(\left(\begin{cases} 1 & \text{for}\: \left(4 a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(2 a + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(2 a + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(6 a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(3 a + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(3 a + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 2*a mod pi = 0\
|| | // 1 for 3*a mod pi = 0\ // 0 for 4*a mod pi = 0\ // 0 for 6*a mod pi = 0\
|| // 0 for 2*a mod pi = 0\ | || | || | || |
|
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\sin{\left(6 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\left(\cot^{2}{\left(2 a \right)} - 1\right) \left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{- \cos{\left(4 a \right)} + 1}{2} & \text{otherwise} \end{cases}\right) & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 3 a \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: 3 a \bmod \pi = 0 \\\cos{\left(6 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 2*a mod pi = 0\ // 1 for 3*a mod pi = 0\
|| | || |
|| 2 | || 2 | // 0 for 4*a mod pi = 0\ // 0 for 6*a mod pi = 0\
|| sin (4*a) | || sin (6*a) | || | || |
||-1 + ----------- | ||-1 + ----------- | || sin(4*a) | || sin(6*a) |
|| 4 | || 4 | ||--------------------------- otherwise | ||--------------------------- otherwise |
|< 4*sin (2*a) |*|< 4*sin (3*a) | - |
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{\sin{\left(4 a \right)}}{\left(1 + \frac{\sin^{2}{\left(4 a \right)}}{4 \sin^{4}{\left(2 a \right)}}\right) \sin^{2}{\left(2 a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\frac{\sin{\left(6 a \right)}}{\left(1 + \frac{\sin^{2}{\left(6 a \right)}}{4 \sin^{4}{\left(3 a \right)}}\right) \sin^{2}{\left(3 a \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(4 a \right)}}{4 \sin^{4}{\left(2 a \right)}}}{1 + \frac{\sin^{2}{\left(4 a \right)}}{4 \sin^{4}{\left(2 a \right)}}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(6 a \right)}}{4 \sin^{4}{\left(3 a \right)}}}{1 + \frac{\sin^{2}{\left(6 a \right)}}{4 \sin^{4}{\left(3 a \right)}}} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 2*a mod pi = 0\
|| | // 1 for 3*a mod pi = 0\ // 0 for 4*a mod pi = 0\ // 0 for 6*a mod pi = 0\
|| // 0 for 2*a mod pi = 0\ | || | || | || |
|| || | | ||/ 1 for 3*a mod pi = 0 | ||/ 0 for 4*a mod pi = 0 | ||/ 0 for 6*a mod pi = 0 |
|| || 2 | | ||| | ||| | ||| |
|
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{2 \cot{\left(2 a \right)}}{\cot^{2}{\left(2 a \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\frac{2 \cot{\left(3 a \right)}}{\cot^{2}{\left(3 a \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\left(\cot^{2}{\left(2 a \right)} - 1\right) \left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(a \right)}}{\left(\cot^{2}{\left(a \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 3 a \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{\cot^{2}{\left(3 a \right)} - 1}{\cot^{2}{\left(3 a \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 2*a mod pi = 0\ // 1 for 3*a mod pi = 0\ || | || | || 2 | || 2 | // 0 for 4*a mod pi = 0\ // 0 for 6*a mod pi = 0\ || cos (2*a) | || cos (3*a) | || | || | ||-1 + -------------- | ||-1 + -------------- | || 2*cos(2*a) | || 2*cos(3*a) | || 2/ pi\ | || 2/ pi\ | ||---------------------------------- otherwise | ||---------------------------------- otherwise | || cos |2*a - --| | || cos |3*a - --| | ||/ 2 \ | ||/ 2 \ | |< \ 2 / |*|< \ 2 / | - |<| cos (2*a) | / pi\ |*|<| cos (3*a) | / pi\ | ||------------------- otherwise | ||------------------- otherwise | |||1 + --------------|*cos|2*a - --| | |||1 + --------------|*cos|3*a - --| | || 2 | || 2 | ||| 2/ pi\| \ 2 / | ||| 2/ pi\| \ 2 / | || cos (2*a) | || cos (3*a) | ||| cos |2*a - --|| | ||| cos |3*a - --|| | || 1 + -------------- | || 1 + -------------- | ||\ \ 2 // | ||\ \ 2 // | || 2/ pi\ | || 2/ pi\ | \\ / \\ / || cos |2*a - --| | || cos |3*a - --| | \\ \ 2 / / \\ \ 2 / /
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{2 \cos{\left(2 a \right)}}{\left(\frac{\cos^{2}{\left(2 a \right)}}{\cos^{2}{\left(2 a - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(2 a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\frac{2 \cos{\left(3 a \right)}}{\left(\frac{\cos^{2}{\left(3 a \right)}}{\cos^{2}{\left(3 a - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(3 a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{\frac{\cos^{2}{\left(2 a \right)}}{\cos^{2}{\left(2 a - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(2 a \right)}}{\cos^{2}{\left(2 a - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{\frac{\cos^{2}{\left(3 a \right)}}{\cos^{2}{\left(3 a - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(3 a \right)}}{\cos^{2}{\left(3 a - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 2*a mod pi = 0\ // 1 for 3*a mod pi = 0\
|| | || | // 0 for 4*a mod pi = 0\ // 0 for 6*a mod pi = 0\
|| 2/ pi\ | || 2/ pi\ | || | || |
|| sec |2*a - --| | || sec |3*a - --| | || / pi\ | || / pi\ |
|| \ 2 / | || \ 2 / | || 2*sec|2*a - --| | || 2*sec|3*a - --| |
||-1 + -------------- | ||-1 + -------------- | || \ 2 / | || \ 2 / |
|| 2 | || 2 | ||----------------------------- otherwise | ||----------------------------- otherwise |
|< sec (2*a) |*|< sec (3*a) | - |
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{2 \sec{\left(2 a - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(2 a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(2 a \right)}}\right) \sec{\left(2 a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\frac{2 \sec{\left(3 a - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(3 a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(3 a \right)}}\right) \sec{\left(3 a \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(2 a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(2 a \right)}}}{1 + \frac{\sec^{2}{\left(2 a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(2 a \right)}}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(3 a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(3 a \right)}}}{1 + \frac{\sec^{2}{\left(3 a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(3 a \right)}}} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 2*a mod pi = 0\ // 1 for 3*a mod pi = 0\ || | || | || 2 | || 2 | // 0 for 4*a mod pi = 0\ // 0 for 6*a mod pi = 0\ || csc (2*a) | || csc (3*a) | || | || | ||-1 + -------------- | ||-1 + -------------- | || 2*csc(2*a) | || 2*csc(3*a) | || 2/pi \ | || 2/pi \ | ||---------------------------------- otherwise | ||---------------------------------- otherwise | || csc |-- - 2*a| | || csc |-- - 3*a| | ||/ 2 \ | ||/ 2 \ | |< \2 / |*|< \2 / | - |<| csc (2*a) | /pi \ |*|<| csc (3*a) | /pi \ | ||------------------- otherwise | ||------------------- otherwise | |||1 + --------------|*csc|-- - 2*a| | |||1 + --------------|*csc|-- - 3*a| | || 2 | || 2 | ||| 2/pi \| \2 / | ||| 2/pi \| \2 / | || csc (2*a) | || csc (3*a) | ||| csc |-- - 2*a|| | ||| csc |-- - 3*a|| | || 1 + -------------- | || 1 + -------------- | ||\ \2 // | ||\ \2 // | || 2/pi \ | || 2/pi \ | \\ / \\ / || csc |-- - 2*a| | || csc |-- - 3*a| | \\ \2 / / \\ \2 / /
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{2 \csc{\left(2 a \right)}}{\left(\frac{\csc^{2}{\left(2 a \right)}}{\csc^{2}{\left(- 2 a + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- 2 a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\frac{2 \csc{\left(3 a \right)}}{\left(\frac{\csc^{2}{\left(3 a \right)}}{\csc^{2}{\left(- 3 a + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- 3 a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{\frac{\csc^{2}{\left(2 a \right)}}{\csc^{2}{\left(- 2 a + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(2 a \right)}}{\csc^{2}{\left(- 2 a + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{\frac{\csc^{2}{\left(3 a \right)}}{\csc^{2}{\left(- 3 a + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(3 a \right)}}{\csc^{2}{\left(- 3 a + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right)$$
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))*Piecewise((1, Mod(3*a = pi, 0)), ((-1 + csc(3*a)^2/csc(pi/2 - 3*a)^2)/(1 + csc(3*a)^2/csc(pi/2 - 3*a)^2), True)) - 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((0, Mod(6*a = pi, 0)), (2*csc(3*a)/((1 + csc(3*a)^2/csc(pi/2 - 3*a)^2)*csc(pi/2 - 3*a)), True))