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sin(10*a)*sin(8*a)+sin(8*a)*sin(6*a) если a=1
Решение
Вы ввели
[src]
sin(10*a)*sin(8*a) + sin(8*a)*sin(6*a)
$$\sin{\left(6 a \right)} \sin{\left(8 a \right)} + \sin{\left(8 a \right)} \sin{\left(10 a \right)}$$
sin(10*a)*sin(8*a) + sin(8*a)*sin(6*a)
Общее упрощение
[src]
(sin(6*a) + sin(10*a))*sin(8*a)
$$\left(\sin{\left(6 a \right)} + \sin{\left(10 a \right)}\right) \sin{\left(8 a \right)}$$
(sin(6*a) + sin(10*a))*sin(8*a)
Подстановка условия
[src]
sin(10*a)*sin(8*a) + sin(8*a)*sin(6*a) при a = 1
подставляем
sin(10*a)*sin(8*a) + sin(8*a)*sin(6*a)
$$\sin{\left(6 a \right)} \sin{\left(8 a \right)} + \sin{\left(8 a \right)} \sin{\left(10 a \right)}$$
(sin(6*a) + sin(10*a))*sin(8*a)
$$\left(\sin{\left(6 a \right)} + \sin{\left(10 a \right)}\right) \sin{\left(8 a \right)}$$
переменные
a = 1
$$a = 1$$
(sin(6*(1)) + sin(10*(1)))*sin(8*(1))
$$\left(\sin{\left(6 (1) \right)} + \sin{\left(10 (1) \right)}\right) \sin{\left(8 (1) \right)}$$
(sin(6*1) + sin(10*1))*sin(8*1)
$$\left(\sin{\left(10 \cdot 1 \right)} + \sin{\left(6 \cdot 1 \right)}\right) \sin{\left(8 \cdot 1 \right)}$$
(sin(6) + sin(10))*sin(8)
$$\left(\sin{\left(10 \right)} + \sin{\left(6 \right)}\right) \sin{\left(8 \right)}$$
(sin(6) + sin(10))*sin(8)
Собрать выражение
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cos(14*a) cos(18*a)
- --------- - --------- + cos(2*a)
2 2
$$\cos{\left(2 a \right)} - \frac{\cos{\left(14 a \right)}}{2} - \frac{\cos{\left(18 a \right)}}{2}$$
-cos(14*a)/2 - cos(18*a)/2 + cos(2*a)
Объединение рациональных выражений
[src]
(sin(6*a) + sin(10*a))*sin(8*a)
$$\left(\sin{\left(6 a \right)} + \sin{\left(10 a \right)}\right) \sin{\left(8 a \right)}$$
(sin(6*a) + sin(10*a))*sin(8*a)
Комбинаторика
[src]
(sin(6*a) + sin(10*a))*sin(8*a)
$$\left(\sin{\left(6 a \right)} + \sin{\left(10 a \right)}\right) \sin{\left(8 a \right)}$$
(sin(6*a) + sin(10*a))*sin(8*a)
Степени
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/ -10*I*a 10*I*a\ / -8*I*a 8*I*a\ / -8*I*a 8*I*a\ / -6*I*a 6*I*a\
\- e + e /*\- e + e / \- e + e /*\- e + e /
- ------------------------------------------- - -----------------------------------------
4 4
$$- \frac{\left(e^{6 i a} - e^{- 6 i a}\right) \left(e^{8 i a} - e^{- 8 i a}\right)}{4} - \frac{\left(e^{8 i a} - e^{- 8 i a}\right) \left(e^{10 i a} - e^{- 10 i a}\right)}{4}$$
-(-exp(-10*i*a) + exp(10*i*a))*(-exp(-8*i*a) + exp(8*i*a))/4 - (-exp(-8*i*a) + exp(8*i*a))*(-exp(-6*i*a) + exp(6*i*a))/4
Тригонометрическая часть
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2 2*sin (8*a)*cos(2*a)
$$2 \sin^{2}{\left(8 a \right)} \cos{\left(2 a \right)}$$
sin(6*a) sin(10*a) -------- + --------- csc(8*a) csc(8*a)
$$\frac{\sin{\left(6 a \right)}}{\csc{\left(8 a \right)}} + \frac{\sin{\left(10 a \right)}}{\csc{\left(8 a \right)}}$$
1 1 ----------------- + ------------------ csc(6*a)*csc(8*a) csc(8*a)*csc(10*a)
$$\frac{1}{\csc{\left(8 a \right)} \csc{\left(10 a \right)}} + \frac{1}{\csc{\left(6 a \right)} \csc{\left(8 a \right)}}$$
/ pi\ / pi\ cos|8*a - --|*sin(6*a) + cos|8*a - --|*sin(10*a) \ 2 / \ 2 /
$$\sin{\left(6 a \right)} \cos{\left(8 a - \frac{\pi}{2} \right)} + \sin{\left(10 a \right)} \cos{\left(8 a - \frac{\pi}{2} \right)}$$
sin(6*a) sin(10*a) ------------- + ------------- csc(pi - 8*a) csc(pi - 8*a)
$$\frac{\sin{\left(6 a \right)}}{\csc{\left(- 8 a + \pi \right)}} + \frac{\sin{\left(10 a \right)}}{\csc{\left(- 8 a + \pi \right)}}$$
/ pi\ / pi\ cos|6*a - --|*sin(8*a) + cos|10*a - --|*sin(8*a) \ 2 / \ 2 /
$$\sin{\left(8 a \right)} \cos{\left(6 a - \frac{\pi}{2} \right)} + \sin{\left(8 a \right)} \cos{\left(10 a - \frac{\pi}{2} \right)}$$
1 1 ---------------------- + ----------------------- csc(6*a)*csc(pi - 8*a) csc(10*a)*csc(pi - 8*a)
$$\frac{1}{\csc{\left(10 a \right)} \csc{\left(- 8 a + \pi \right)}} + \frac{1}{\csc{\left(6 a \right)} \csc{\left(- 8 a + \pi \right)}}$$
sin(6*a) sin(10*a) ------------- + ------------- / pi\ / pi\ sec|8*a - --| sec|8*a - --| \ 2 / \ 2 /
$$\frac{\sin{\left(6 a \right)}}{\sec{\left(8 a - \frac{\pi}{2} \right)}} + \frac{\sin{\left(10 a \right)}}{\sec{\left(8 a - \frac{\pi}{2} \right)}}$$
1 1 ---------------------------- + --------------------------- csc(pi - 10*a)*csc(pi - 8*a) csc(pi - 8*a)*csc(pi - 6*a)
$$\frac{1}{\csc{\left(- 8 a + \pi \right)} \csc{\left(- 6 a + \pi \right)}} + \frac{1}{\csc{\left(- 10 a + \pi \right)} \csc{\left(- 8 a + \pi \right)}}$$
sin(6*a) sin(10*a) ------------- + ------------- /pi \ /pi \ sec|-- - 8*a| sec|-- - 8*a| \2 / \2 /
$$\frac{\sin{\left(6 a \right)}}{\sec{\left(- 8 a + \frac{\pi}{2} \right)}} + \frac{\sin{\left(10 a \right)}}{\sec{\left(- 8 a + \frac{\pi}{2} \right)}}$$
1 1
---------------------- + -----------------------
/ pi\ / pi\
csc(8*a)*sec|6*a - --| csc(8*a)*sec|10*a - --|
\ 2 / \ 2 /
$$\frac{1}{\csc{\left(8 a \right)} \sec{\left(10 a - \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(8 a \right)} \sec{\left(6 a - \frac{\pi}{2} \right)}}$$
/ pi\ / pi\ / pi\ / pi\ cos|6*a - --|*cos|8*a - --| + cos|8*a - --|*cos|10*a - --| \ 2 / \ 2 / \ 2 / \ 2 /
$$\cos{\left(6 a - \frac{\pi}{2} \right)} \cos{\left(8 a - \frac{\pi}{2} \right)} + \cos{\left(8 a - \frac{\pi}{2} \right)} \cos{\left(10 a - \frac{\pi}{2} \right)}$$
1 1 --------------------------- + ---------------------------- / pi\ / pi\ / pi\ / pi\ sec|6*a - --|*sec|8*a - --| sec|8*a - --|*sec|10*a - --| \ 2 / \ 2 / \ 2 / \ 2 /
$$\frac{1}{\sec{\left(8 a - \frac{\pi}{2} \right)} \sec{\left(10 a - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(6 a - \frac{\pi}{2} \right)} \sec{\left(8 a - \frac{\pi}{2} \right)}}$$
1 1 --------------------------- + ---------------------------- /pi \ / pi\ /pi \ / pi\ sec|-- - 8*a|*sec|6*a - --| sec|-- - 8*a|*sec|10*a - --| \2 / \ 2 / \2 / \ 2 /
$$\frac{1}{\sec{\left(- 8 a + \frac{\pi}{2} \right)} \sec{\left(10 a - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(- 8 a + \frac{\pi}{2} \right)} \sec{\left(6 a - \frac{\pi}{2} \right)}}$$
1 1 ---------------------------- + --------------------------- /pi \ /pi \ /pi \ /pi \ sec|-- - 10*a|*sec|-- - 8*a| sec|-- - 8*a|*sec|-- - 6*a| \2 / \2 / \2 / \2 /
$$\frac{1}{\sec{\left(- 8 a + \frac{\pi}{2} \right)} \sec{\left(- 6 a + \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(- 10 a + \frac{\pi}{2} \right)} \sec{\left(- 8 a + \frac{\pi}{2} \right)}}$$
4*cos(3*a)*cos(4*a)*sin(3*a)*sin(4*a) + 4*cos(4*a)*cos(5*a)*sin(4*a)*sin(5*a)
$$4 \sin{\left(3 a \right)} \sin{\left(4 a \right)} \cos{\left(3 a \right)} \cos{\left(4 a \right)} + 4 \sin{\left(4 a \right)} \sin{\left(5 a \right)} \cos{\left(4 a \right)} \cos{\left(5 a \right)}$$
4*cos(3*a)*sin(3*a)*sin(4*a) 4*cos(5*a)*sin(4*a)*sin(5*a)
---------------------------- + ----------------------------
sec(4*a) sec(4*a)
$$\frac{4 \sin{\left(3 a \right)} \sin{\left(4 a \right)} \cos{\left(3 a \right)}}{\sec{\left(4 a \right)}} + \frac{4 \sin{\left(4 a \right)} \sin{\left(5 a \right)} \cos{\left(5 a \right)}}{\sec{\left(4 a \right)}}$$
2 2 2 2 4*cos (3*a)*cos (4*a)*tan(3*a)*tan(4*a) + 4*cos (4*a)*cos (5*a)*tan(4*a)*tan(5*a)
$$4 \cos^{2}{\left(3 a \right)} \cos^{2}{\left(4 a \right)} \tan{\left(3 a \right)} \tan{\left(4 a \right)} + 4 \cos^{2}{\left(4 a \right)} \cos^{2}{\left(5 a \right)} \tan{\left(4 a \right)} \tan{\left(5 a \right)}$$
2 2 2 2 4*sin (3*a)*sin (4*a)*cot(3*a)*cot(4*a) + 4*sin (4*a)*sin (5*a)*cot(4*a)*cot(5*a)
$$4 \sin^{2}{\left(3 a \right)} \sin^{2}{\left(4 a \right)} \cot{\left(3 a \right)} \cot{\left(4 a \right)} + 4 \sin^{2}{\left(4 a \right)} \sin^{2}{\left(5 a \right)} \cot{\left(4 a \right)} \cot{\left(5 a \right)}$$
4*cos(3*a)*sin(3*a)*sin(4*a) 4*cos(5*a)*sin(4*a)*sin(5*a)
---------------------------- + ----------------------------
/pi \ /pi \
csc|-- - 4*a| csc|-- - 4*a|
\2 / \2 /
$$\frac{4 \sin{\left(3 a \right)} \sin{\left(4 a \right)} \cos{\left(3 a \right)}}{\csc{\left(- 4 a + \frac{\pi}{2} \right)}} + \frac{4 \sin{\left(4 a \right)} \sin{\left(5 a \right)} \cos{\left(5 a \right)}}{\csc{\left(- 4 a + \frac{\pi}{2} \right)}}$$
/ pi\ / pi\ / pi\ / pi\
4*cos(3*a)*cos(4*a)*cos|3*a - --|*cos|4*a - --| + 4*cos(4*a)*cos(5*a)*cos|4*a - --|*cos|5*a - --|
\ 2 / \ 2 / \ 2 / \ 2 /
$$4 \cos{\left(3 a \right)} \cos{\left(4 a \right)} \cos{\left(3 a - \frac{\pi}{2} \right)} \cos{\left(4 a - \frac{\pi}{2} \right)} + 4 \cos{\left(4 a \right)} \cos{\left(5 a \right)} \cos{\left(4 a - \frac{\pi}{2} \right)} \cos{\left(5 a - \frac{\pi}{2} \right)}$$
4 4
--------------------------------------------- + ---------------------------------------------
/ pi\ / pi\ / pi\ / pi\
sec(3*a)*sec(4*a)*sec|3*a - --|*sec|4*a - --| sec(4*a)*sec(5*a)*sec|4*a - --|*sec|5*a - --|
\ 2 / \ 2 / \ 2 / \ 2 /
$$\frac{4}{\sec{\left(4 a \right)} \sec{\left(5 a \right)} \sec{\left(4 a - \frac{\pi}{2} \right)} \sec{\left(5 a - \frac{\pi}{2} \right)}} + \frac{4}{\sec{\left(3 a \right)} \sec{\left(4 a \right)} \sec{\left(3 a - \frac{\pi}{2} \right)} \sec{\left(4 a - \frac{\pi}{2} \right)}}$$
4 4
--------------------------------------------- + ---------------------------------------------
/pi \ /pi \ /pi \ /pi \
csc(3*a)*csc(4*a)*csc|-- - 4*a|*csc|-- - 3*a| csc(4*a)*csc(5*a)*csc|-- - 5*a|*csc|-- - 4*a|
\2 / \2 / \2 / \2 /
$$\frac{4}{\csc{\left(4 a \right)} \csc{\left(5 a \right)} \csc{\left(- 5 a + \frac{\pi}{2} \right)} \csc{\left(- 4 a + \frac{\pi}{2} \right)}} + \frac{4}{\csc{\left(3 a \right)} \csc{\left(4 a \right)} \csc{\left(- 4 a + \frac{\pi}{2} \right)} \csc{\left(- 3 a + \frac{\pi}{2} \right)}}$$
2 /pi \ 2 /pi \
4*cos (4*a)*cos(3*a)*csc|-- - 4*a|*sin(3*a)*sin(4*a) + 4*cos (4*a)*cos(5*a)*csc|-- - 4*a|*sin(4*a)*sin(5*a)
\2 / \2 /
$$4 \sin{\left(3 a \right)} \sin{\left(4 a \right)} \cos{\left(3 a \right)} \cos^{2}{\left(4 a \right)} \csc{\left(- 4 a + \frac{\pi}{2} \right)} + 4 \sin{\left(4 a \right)} \sin{\left(5 a \right)} \cos^{2}{\left(4 a \right)} \cos{\left(5 a \right)} \csc{\left(- 4 a + \frac{\pi}{2} \right)}$$
4*tan(3*a)*tan(4*a) 4*tan(4*a)*tan(5*a) ------------------------------- + ------------------------------- / 2 \ / 2 \ / 2 \ / 2 \ \1 + tan (3*a)/*\1 + tan (4*a)/ \1 + tan (4*a)/*\1 + tan (5*a)/
$$\frac{4 \tan{\left(4 a \right)} \tan{\left(5 a \right)}}{\left(\tan^{2}{\left(4 a \right)} + 1\right) \left(\tan^{2}{\left(5 a \right)} + 1\right)} + \frac{4 \tan{\left(3 a \right)} \tan{\left(4 a \right)}}{\left(\tan^{2}{\left(3 a \right)} + 1\right) \left(\tan^{2}{\left(4 a \right)} + 1\right)}$$
4*cot(3*a)*cot(4*a) 4*cot(4*a)*cot(5*a) ------------------------------- + ------------------------------- / 2 \ / 2 \ / 2 \ / 2 \ \1 + cot (3*a)/*\1 + cot (4*a)/ \1 + cot (4*a)/*\1 + cot (5*a)/
$$\frac{4 \cot{\left(4 a \right)} \cot{\left(5 a \right)}}{\left(\cot^{2}{\left(4 a \right)} + 1\right) \left(\cot^{2}{\left(5 a \right)} + 1\right)} + \frac{4 \cot{\left(3 a \right)} \cot{\left(4 a \right)}}{\left(\cot^{2}{\left(3 a \right)} + 1\right) \left(\cot^{2}{\left(4 a \right)} + 1\right)}$$
2 2 2 2 4*cos (3*a)*cos (4*a)*sec(3*a)*sec(4*a)*sin(3*a)*sin(4*a) + 4*cos (4*a)*cos (5*a)*sec(4*a)*sec(5*a)*sin(4*a)*sin(5*a)
$$4 \sin{\left(3 a \right)} \sin{\left(4 a \right)} \cos^{2}{\left(3 a \right)} \cos^{2}{\left(4 a \right)} \sec{\left(3 a \right)} \sec{\left(4 a \right)} + 4 \sin{\left(4 a \right)} \sin{\left(5 a \right)} \cos^{2}{\left(4 a \right)} \cos^{2}{\left(5 a \right)} \sec{\left(4 a \right)} \sec{\left(5 a \right)}$$
2 2/ pi\ / 2/ pi\\ 2 2/ pi\ / 2/ pi\\
2*cos (3*a)*sin |4*a + --|*|1 - cot |4*a + --||*tan(3*a) + 2*cos (5*a)*sin |4*a + --|*|1 - cot |4*a + --||*tan(5*a)
\ 4 / \ \ 4 // \ 4 / \ \ 4 //
$$2 \cdot \left(- \cot^{2}{\left(4 a + \frac{\pi}{4} \right)} + 1\right) \sin^{2}{\left(4 a + \frac{\pi}{4} \right)} \cos^{2}{\left(3 a \right)} \tan{\left(3 a \right)} + 2 \cdot \left(- \cot^{2}{\left(4 a + \frac{\pi}{4} \right)} + 1\right) \sin^{2}{\left(4 a + \frac{\pi}{4} \right)} \cos^{2}{\left(5 a \right)} \tan{\left(5 a \right)}$$
4 4 ------------------------------------------------- + ------------------------------------------------- / 1 \ / 1 \ / 1 \ / 1 \ |1 + ---------|*|1 + ---------|*cot(3*a)*cot(4*a) |1 + ---------|*|1 + ---------|*cot(4*a)*cot(5*a) | 2 | | 2 | | 2 | | 2 | \ cot (3*a)/ \ cot (4*a)/ \ cot (4*a)/ \ cot (5*a)/
$$\frac{4}{\left(1 + \frac{1}{\cot^{2}{\left(4 a \right)}}\right) \left(1 + \frac{1}{\cot^{2}{\left(5 a \right)}}\right) \cot{\left(4 a \right)} \cot{\left(5 a \right)}} + \frac{4}{\left(1 + \frac{1}{\cot^{2}{\left(3 a \right)}}\right) \left(1 + \frac{1}{\cot^{2}{\left(4 a \right)}}\right) \cot{\left(3 a \right)} \cot{\left(4 a \right)}}$$
// 0 for 6*a mod pi = 0\ // 0 for 10*a mod pi = 0\ 2*|< |*cos(4*a)*sin(4*a) + 2*|< |*cos(4*a)*sin(4*a) \\sin(6*a) otherwise / \\sin(10*a) otherwise /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\sin{\left(6 a \right)} & \text{otherwise} \end{cases}\right) \sin{\left(4 a \right)} \cos{\left(4 a \right)}\right) + \left(2 \left(\begin{cases} 0 & \text{for}\: 10 a \bmod \pi = 0 \\\sin{\left(10 a \right)} & \text{otherwise} \end{cases}\right) \sin{\left(4 a \right)} \cos{\left(4 a \right)}\right)$$
2 // 0 for 6*a mod pi = 0\ 2 // 0 for 10*a mod pi = 0\
2*sin (4*a)*|< |*cot(4*a) + 2*sin (4*a)*|< |*cot(4*a)
\\sin(6*a) otherwise / \\sin(10*a) otherwise /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\sin{\left(6 a \right)} & \text{otherwise} \end{cases}\right) \sin^{2}{\left(4 a \right)} \cot{\left(4 a \right)}\right) + \left(2 \left(\begin{cases} 0 & \text{for}\: 10 a \bmod \pi = 0 \\\sin{\left(10 a \right)} & \text{otherwise} \end{cases}\right) \sin^{2}{\left(4 a \right)} \cot{\left(4 a \right)}\right)$$
2 2 2 2/pi \ 2 2 2 2/pi \
16*cos (3*a)*sin (3*a)*sin (4*a)*sin |-- + 4*a| 16*cos (5*a)*sin (4*a)*sin (5*a)*sin |-- + 4*a|
\2 / \2 /
----------------------------------------------- + -----------------------------------------------
sin(6*a)*sin(8*a) sin(8*a)*sin(10*a)
$$\frac{16 \sin^{2}{\left(3 a \right)} \sin^{2}{\left(4 a \right)} \sin^{2}{\left(4 a + \frac{\pi}{2} \right)} \cos^{2}{\left(3 a \right)}}{\sin{\left(6 a \right)} \sin{\left(8 a \right)}} + \frac{16 \sin^{2}{\left(4 a \right)} \sin^{2}{\left(5 a \right)} \sin^{2}{\left(4 a + \frac{\pi}{2} \right)} \cos^{2}{\left(5 a \right)}}{\sin{\left(8 a \right)} \sin{\left(10 a \right)}}$$
2 2 2/pi \ 2/pi \ 2 2 2/pi \ 2/pi \
16*sin (3*a)*sin (4*a)*sin |-- + 3*a|*sin |-- + 4*a| 16*sin (4*a)*sin (5*a)*sin |-- + 4*a|*sin |-- + 5*a|
\2 / \2 / \2 / \2 /
---------------------------------------------------- + ----------------------------------------------------
sin(6*a)*sin(8*a) sin(8*a)*sin(10*a)
$$\frac{16 \sin^{2}{\left(3 a \right)} \sin^{2}{\left(4 a \right)} \sin^{2}{\left(3 a + \frac{\pi}{2} \right)} \sin^{2}{\left(4 a + \frac{\pi}{2} \right)}}{\sin{\left(6 a \right)} \sin{\left(8 a \right)}} + \frac{16 \sin^{2}{\left(4 a \right)} \sin^{2}{\left(5 a \right)} \sin^{2}{\left(4 a + \frac{\pi}{2} \right)} \sin^{2}{\left(5 a + \frac{\pi}{2} \right)}}{\sin{\left(8 a \right)} \sin{\left(10 a \right)}}$$
/ 6 2 4 \ / 6 2 4 \ 16*\1 - 16*sin (a) - 10*sin (a) + 24*sin (a)/*cos(a)*cos(3*a)*sin(a)*sin(3*a) + 16*\1 - 16*sin (a) - 10*sin (a) + 24*sin (a)/*cos(a)*cos(5*a)*sin(a)*sin(5*a)
$$16 \left(- 16 \sin^{6}{\left(a \right)} + 24 \sin^{4}{\left(a \right)} - 10 \sin^{2}{\left(a \right)} + 1\right) \sin{\left(a \right)} \sin{\left(3 a \right)} \cos{\left(a \right)} \cos{\left(3 a \right)} + 16 \left(- 16 \sin^{6}{\left(a \right)} + 24 \sin^{4}{\left(a \right)} - 10 \sin^{2}{\left(a \right)} + 1\right) \sin{\left(a \right)} \sin{\left(5 a \right)} \cos{\left(a \right)} \cos{\left(5 a \right)}$$
/ 2/ pi\\ / 2/ pi\\
2*|1 - cot |4*a + --||*tan(3*a) 2*|1 - cot |4*a + --||*tan(5*a)
\ \ 4 // \ \ 4 //
------------------------------------ + ------------------------------------
/ 2/ pi\\ / 2 \ / 2/ pi\\ / 2 \
|1 + cot |4*a + --||*\1 + tan (3*a)/ |1 + cot |4*a + --||*\1 + tan (5*a)/
\ \ 4 // \ \ 4 //
$$\frac{2 \cdot \left(- \cot^{2}{\left(4 a + \frac{\pi}{4} \right)} + 1\right) \tan{\left(5 a \right)}}{\left(\tan^{2}{\left(5 a \right)} + 1\right) \left(\cot^{2}{\left(4 a + \frac{\pi}{4} \right)} + 1\right)} + \frac{2 \cdot \left(- \cot^{2}{\left(4 a + \frac{\pi}{4} \right)} + 1\right) \tan{\left(3 a \right)}}{\left(\tan^{2}{\left(3 a \right)} + 1\right) \left(\cot^{2}{\left(4 a + \frac{\pi}{4} \right)} + 1\right)}$$
2 2 2 / 2 2 \ (1 + cos(4*a)) *(1 + tan(2*a)) *(-1 + tan(2*a)) *\-tan(3*a) - tan(5*a) + 2*sin(3*a) + 2*sin(5*a) + (1 - cos(3*a)) *tan(3*a) + (1 - cos(5*a)) *tan(5*a)/*tan(4*a)
$$\left(\cos{\left(4 a \right)} + 1\right)^{2} \left(\tan{\left(2 a \right)} - 1\right)^{2} \left(\tan{\left(2 a \right)} + 1\right)^{2} \left(\left(- \cos{\left(3 a \right)} + 1\right)^{2} \tan{\left(3 a \right)} + \left(- \cos{\left(5 a \right)} + 1\right)^{2} \tan{\left(5 a \right)} + 2 \sin{\left(3 a \right)} + 2 \sin{\left(5 a \right)} - \tan{\left(3 a \right)} - \tan{\left(5 a \right)}\right) \tan{\left(4 a \right)}$$
2/ pi\ / 2/ pi\\ / 2/ pi\\ 2/ pi\ / 2/ pi\\ / 2/ pi\\
sin |4*a + --|*|1 - cot |3*a + --||*|1 - cot |4*a + --||*(1 + sin(6*a)) sin |4*a + --|*|1 - cot |4*a + --||*|1 - cot |5*a + --||*(1 + sin(10*a))
\ 4 / \ \ 4 // \ \ 4 // \ 4 / \ \ 4 // \ \ 4 //
----------------------------------------------------------------------- + ------------------------------------------------------------------------
2 2
$$\frac{\left(- \cot^{2}{\left(3 a + \frac{\pi}{4} \right)} + 1\right) \left(- \cot^{2}{\left(4 a + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(6 a \right)} + 1\right) \sin^{2}{\left(4 a + \frac{\pi}{4} \right)}}{2} + \frac{\left(- \cot^{2}{\left(4 a + \frac{\pi}{4} \right)} + 1\right) \left(- \cot^{2}{\left(5 a + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(10 a \right)} + 1\right) \sin^{2}{\left(4 a + \frac{\pi}{4} \right)}}{2}$$
// 0 for 6*a mod pi = 0\ // 0 for 10*a mod pi = 0\
|| | || |
|| 2*cot(3*a) | || 2*cot(5*a) |
2*|<------------- otherwise |*cot(4*a) 2*|<------------- otherwise |*cot(4*a)
|| 2 | || 2 |
||1 + cot (3*a) | ||1 + cot (5*a) |
\\ / \\ /
----------------------------------------------- + ------------------------------------------------
2 2
1 + cot (4*a) 1 + cot (4*a)
$$\left(\frac{2 \left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\frac{2 \cot{\left(3 a \right)}}{\cot^{2}{\left(3 a \right)} + 1} & \text{otherwise} \end{cases}\right) \cot{\left(4 a \right)}}{\cot^{2}{\left(4 a \right)} + 1}\right) + \left(\frac{2 \left(\begin{cases} 0 & \text{for}\: 10 a \bmod \pi = 0 \\\frac{2 \cot{\left(5 a \right)}}{\cot^{2}{\left(5 a \right)} + 1} & \text{otherwise} \end{cases}\right) \cot{\left(4 a \right)}}{\cot^{2}{\left(4 a \right)} + 1}\right)$$
// 0 for 6*a mod pi = 0\ // 0 for 8*a mod pi = 0\ // 0 for 8*a mod pi = 0\ // 0 for 10*a mod pi = 0\ |< |*|< | + |< |*|< | \\sin(6*a) otherwise / \\sin(8*a) otherwise / \\sin(8*a) otherwise / \\sin(10*a) otherwise /
$$\left(\left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\sin{\left(6 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 8 a \bmod \pi = 0 \\\sin{\left(8 a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 0 & \text{for}\: 8 a \bmod \pi = 0 \\\sin{\left(8 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 10 a \bmod \pi = 0 \\\sin{\left(10 a \right)} & \text{otherwise} \end{cases}\right)\right)$$
/ 2/ pi\\ / 2/ pi\\ / 2/ pi\\ / 2/ pi\\ |-1 + tan |3*a + --||*|-1 + tan |4*a + --|| |-1 + tan |4*a + --||*|-1 + tan |5*a + --|| \ \ 4 // \ \ 4 // \ \ 4 // \ \ 4 // ------------------------------------------- + ------------------------------------------- / 2/ pi\\ / 2/ pi\\ / 2/ pi\\ / 2/ pi\\ |1 + tan |3*a + --||*|1 + tan |4*a + --|| |1 + tan |4*a + --||*|1 + tan |5*a + --|| \ \ 4 // \ \ 4 // \ \ 4 // \ \ 4 //
$$\frac{\left(\tan^{2}{\left(3 a + \frac{\pi}{4} \right)} - 1\right) \left(\tan^{2}{\left(4 a + \frac{\pi}{4} \right)} - 1\right)}{\left(\tan^{2}{\left(3 a + \frac{\pi}{4} \right)} + 1\right) \left(\tan^{2}{\left(4 a + \frac{\pi}{4} \right)} + 1\right)} + \frac{\left(\tan^{2}{\left(4 a + \frac{\pi}{4} \right)} - 1\right) \left(\tan^{2}{\left(5 a + \frac{\pi}{4} \right)} - 1\right)}{\left(\tan^{2}{\left(4 a + \frac{\pi}{4} \right)} + 1\right) \left(\tan^{2}{\left(5 a + \frac{\pi}{4} \right)} + 1\right)}$$
/ 2/ pi\\ / 2/ pi\\ / 2/ pi\\ / 2/ pi\\ |1 - cot |3*a + --||*|1 - cot |4*a + --|| |1 - cot |4*a + --||*|1 - cot |5*a + --|| \ \ 4 // \ \ 4 // \ \ 4 // \ \ 4 // ----------------------------------------- + ----------------------------------------- / 2/ pi\\ / 2/ pi\\ / 2/ pi\\ / 2/ pi\\ |1 + cot |3*a + --||*|1 + cot |4*a + --|| |1 + cot |4*a + --||*|1 + cot |5*a + --|| \ \ 4 // \ \ 4 // \ \ 4 // \ \ 4 //
$$\frac{\left(- \cot^{2}{\left(3 a + \frac{\pi}{4} \right)} + 1\right) \left(- \cot^{2}{\left(4 a + \frac{\pi}{4} \right)} + 1\right)}{\left(\cot^{2}{\left(3 a + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(4 a + \frac{\pi}{4} \right)} + 1\right)} + \frac{\left(- \cot^{2}{\left(4 a + \frac{\pi}{4} \right)} + 1\right) \left(- \cot^{2}{\left(5 a + \frac{\pi}{4} \right)} + 1\right)}{\left(\cot^{2}{\left(4 a + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(5 a + \frac{\pi}{4} \right)} + 1\right)}$$
// 0 for 6*a mod pi = 0\ // 0 for 8*a mod pi = 0\ // 0 for 8*a mod pi = 0\ // 0 for 10*a mod pi = 0\ || | || | || | || | |< 1 |*|< 1 | + |< 1 |*|< 1 | ||-------- otherwise | ||-------- otherwise | ||-------- otherwise | ||--------- otherwise | \\csc(6*a) / \\csc(8*a) / \\csc(8*a) / \\csc(10*a) /
$$\left(\left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\frac{1}{\csc{\left(6 a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 8 a \bmod \pi = 0 \\\frac{1}{\csc{\left(8 a \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 0 & \text{for}\: 8 a \bmod \pi = 0 \\\frac{1}{\csc{\left(8 a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 10 a \bmod \pi = 0 \\\frac{1}{\csc{\left(10 a \right)}} & \text{otherwise} \end{cases}\right)\right)$$
2 2 2 2
/ 2 \ / 2/3*a\\ / 2 \ / 2/5*a\\
4*\1 - tan (2*a)/ *|1 - tan |---|| *tan(3*a)*tan(4*a) 4*\1 - tan (2*a)/ *|1 - tan |---|| *tan(4*a)*tan(5*a)
\ \ 2 // \ \ 2 //
----------------------------------------------------- + -----------------------------------------------------
2 2 2 2
/ 2 \ / 2/3*a\\ / 2 \ / 2/5*a\\
\1 + tan (2*a)/ *|1 + tan |---|| \1 + tan (2*a)/ *|1 + tan |---||
\ \ 2 // \ \ 2 //
$$\frac{4 \left(- \tan^{2}{\left(\frac{3 a}{2} \right)} + 1\right)^{2} \left(- \tan^{2}{\left(2 a \right)} + 1\right)^{2} \tan{\left(3 a \right)} \tan{\left(4 a \right)}}{\left(\tan^{2}{\left(\frac{3 a}{2} \right)} + 1\right)^{2} \left(\tan^{2}{\left(2 a \right)} + 1\right)^{2}} + \frac{4 \left(- \tan^{2}{\left(2 a \right)} + 1\right)^{2} \left(- \tan^{2}{\left(\frac{5 a}{2} \right)} + 1\right)^{2} \tan{\left(4 a \right)} \tan{\left(5 a \right)}}{\left(\tan^{2}{\left(2 a \right)} + 1\right)^{2} \left(\tan^{2}{\left(\frac{5 a}{2} \right)} + 1\right)^{2}}$$
2 2 2 2
16*sin (3*a)*sin (4*a) 16*sin (4*a)*sin (5*a)
----------------------------------------------------- + ------------------------------------------------------
/ 4 \ / 4 \ / 4 \ / 4 \
| 4*sin (3*a)| | 4*sin (4*a)| | 4*sin (4*a)| | 4*sin (5*a)|
|1 + -----------|*|1 + -----------|*sin(6*a)*sin(8*a) |1 + -----------|*|1 + -----------|*sin(8*a)*sin(10*a)
| 2 | | 2 | | 2 | | 2 |
\ sin (6*a) / \ sin (8*a) / \ sin (8*a) / \ sin (10*a)/
$$\frac{16 \sin^{2}{\left(4 a \right)} \sin^{2}{\left(5 a \right)}}{\left(\frac{4 \sin^{4}{\left(4 a \right)}}{\sin^{2}{\left(8 a \right)}} + 1\right) \left(\frac{4 \sin^{4}{\left(5 a \right)}}{\sin^{2}{\left(10 a \right)}} + 1\right) \sin{\left(8 a \right)} \sin{\left(10 a \right)}} + \frac{16 \sin^{2}{\left(3 a \right)} \sin^{2}{\left(4 a \right)}}{\left(\frac{4 \sin^{4}{\left(3 a \right)}}{\sin^{2}{\left(6 a \right)}} + 1\right) \left(\frac{4 \sin^{4}{\left(4 a \right)}}{\sin^{2}{\left(8 a \right)}} + 1\right) \sin{\left(6 a \right)} \sin{\left(8 a \right)}}$$
// 0 for 6*a mod pi = 0\ // 0 for 8*a mod pi = 0\ // 0 for 8*a mod pi = 0\ // 0 for 10*a mod pi = 0\ || | || | || | || | |< / pi\ |*|< / pi\ | + |< / pi\ |*|< / pi\ | ||cos|6*a - --| otherwise | ||cos|8*a - --| otherwise | ||cos|8*a - --| otherwise | ||cos|10*a - --| otherwise | \\ \ 2 / / \\ \ 2 / / \\ \ 2 / / \\ \ 2 / /
$$\left(\left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\cos{\left(6 a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 8 a \bmod \pi = 0 \\\cos{\left(8 a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 0 & \text{for}\: 8 a \bmod \pi = 0 \\\cos{\left(8 a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 10 a \bmod \pi = 0 \\\cos{\left(10 a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for 6*a mod pi = 0\ // 0 for 10*a mod pi = 0\
|| | || |
/ 2/ pi\\ || 2*cot(3*a) | / 2/ pi\\ || 2*cot(5*a) |
|-1 + tan |4*a + --||*|<------------- otherwise | |-1 + tan |4*a + --||*|<------------- otherwise |
\ \ 4 // || 2 | \ \ 4 // || 2 |
||1 + cot (3*a) | ||1 + cot (5*a) |
\\ / \\ /
---------------------------------------------------------- + -----------------------------------------------------------
2/ pi\ 2/ pi\
1 + tan |4*a + --| 1 + tan |4*a + --|
\ 4 / \ 4 /
$$\left(\frac{\left(\tan^{2}{\left(4 a + \frac{\pi}{4} \right)} - 1\right) \left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\frac{2 \cot{\left(3 a \right)}}{\cot^{2}{\left(3 a \right)} + 1} & \text{otherwise} \end{cases}\right)}{\tan^{2}{\left(4 a + \frac{\pi}{4} \right)} + 1}\right) + \left(\frac{\left(\tan^{2}{\left(4 a + \frac{\pi}{4} \right)} - 1\right) \left(\begin{cases} 0 & \text{for}\: 10 a \bmod \pi = 0 \\\frac{2 \cot{\left(5 a \right)}}{\cot^{2}{\left(5 a \right)} + 1} & \text{otherwise} \end{cases}\right)}{\tan^{2}{\left(4 a + \frac{\pi}{4} \right)} + 1}\right)$$
// 0 for 6*a mod pi = 0\ // 0 for 8*a mod pi = 0\ // 0 for 8*a mod pi = 0\ // 0 for 10*a mod pi = 0\ || | || | || | || | || 1 | || 1 | || 1 | || 1 | |<------------- otherwise |*|<------------- otherwise | + |<------------- otherwise |*|<-------------- otherwise | || / pi\ | || / pi\ | || / pi\ | || / pi\ | ||sec|6*a - --| | ||sec|8*a - --| | ||sec|8*a - --| | ||sec|10*a - --| | \\ \ 2 / / \\ \ 2 / / \\ \ 2 / / \\ \ 2 / /
$$\left(\left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\frac{1}{\sec{\left(6 a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 8 a \bmod \pi = 0 \\\frac{1}{\sec{\left(8 a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 0 & \text{for}\: 8 a \bmod \pi = 0 \\\frac{1}{\sec{\left(8 a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 10 a \bmod \pi = 0 \\\frac{1}{\sec{\left(10 a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right)$$
/ 2 \ / 2 \ |/ 2 2 \ 2 2 | / 2 2 \ |/ 2 2 \ 2 2 | / 2 2 \ 16*\\cos (a) - sin (a)/ - 4*cos (a)*sin (a)/*\cos (a) - sin (a)/*cos(a)*cos(3*a)*sin(a)*sin(3*a) + 16*\\cos (a) - sin (a)/ - 4*cos (a)*sin (a)/*\cos (a) - sin (a)/*cos(a)*cos(5*a)*sin(a)*sin(5*a)
$$16 \left(- 4 \sin^{2}{\left(a \right)} \cos^{2}{\left(a \right)} + \left(- \sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}\right)^{2}\right) \left(- \sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}\right) \sin{\left(a \right)} \sin{\left(3 a \right)} \cos{\left(a \right)} \cos{\left(3 a \right)} + 16 \left(- 4 \sin^{2}{\left(a \right)} \cos^{2}{\left(a \right)} + \left(- \sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}\right)^{2}\right) \left(- \sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}\right) \sin{\left(a \right)} \sin{\left(5 a \right)} \cos{\left(a \right)} \cos{\left(5 a \right)}$$
// 1 for 2*a mod pi = 0\ // 1 for 3*a mod 2*pi = 0\ // 1 for 2*a mod pi = 0\ // 1 for 5*a mod 2*pi = 0\ || | || | || | || | 4*|<1 + cos(8*a) |*|< 2 |*tan(3*a)*tan(4*a) + 4*|<1 + cos(8*a) |*|< 2 |*tan(4*a)*tan(5*a) ||------------ otherwise | ||cos (3*a) otherwise | ||------------ otherwise | ||cos (5*a) otherwise | \\ 2 / \\ / \\ 2 / \\ /
$$\left(4 \left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{\cos{\left(8 a \right)} + 1}{2} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\cos^{2}{\left(3 a \right)} & \text{otherwise} \end{cases}\right) \tan{\left(3 a \right)} \tan{\left(4 a \right)}\right) + \left(4 \left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{\cos{\left(8 a \right)} + 1}{2} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 5 a \bmod 2 \pi = 0 \\\cos^{2}{\left(5 a \right)} & \text{otherwise} \end{cases}\right) \tan{\left(4 a \right)} \tan{\left(5 a \right)}\right)$$
4*sec(3*a)*sec(4*a) 4*sec(4*a)*sec(5*a) --------------------------------------------------------------------- + --------------------------------------------------------------------- / 2 \ / 2 \ / 2 \ / 2 \ | sec (3*a) | | sec (4*a) | / pi\ / pi\ | sec (4*a) | | sec (5*a) | / pi\ / pi\ |1 + --------------|*|1 + --------------|*sec|3*a - --|*sec|4*a - --| |1 + --------------|*|1 + --------------|*sec|4*a - --|*sec|5*a - --| | 2/ pi\| | 2/ pi\| \ 2 / \ 2 / | 2/ pi\| | 2/ pi\| \ 2 / \ 2 / | sec |3*a - --|| | sec |4*a - --|| | sec |4*a - --|| | sec |5*a - --|| \ \ 2 // \ \ 2 // \ \ 2 // \ \ 2 //
$$\frac{4 \sec{\left(4 a \right)} \sec{\left(5 a \right)}}{\left(\frac{\sec^{2}{\left(4 a \right)}}{\sec^{2}{\left(4 a - \frac{\pi}{2} \right)}} + 1\right) \left(\frac{\sec^{2}{\left(5 a \right)}}{\sec^{2}{\left(5 a - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(4 a - \frac{\pi}{2} \right)} \sec{\left(5 a - \frac{\pi}{2} \right)}} + \frac{4 \sec{\left(3 a \right)} \sec{\left(4 a \right)}}{\left(\frac{\sec^{2}{\left(3 a \right)}}{\sec^{2}{\left(3 a - \frac{\pi}{2} \right)}} + 1\right) \left(\frac{\sec^{2}{\left(4 a \right)}}{\sec^{2}{\left(4 a - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(3 a - \frac{\pi}{2} \right)} \sec{\left(4 a - \frac{\pi}{2} \right)}}$$
/ pi\ / pi\ / pi\ / pi\
4*cos|3*a - --|*cos|4*a - --| 4*cos|4*a - --|*cos|5*a - --|
\ 2 / \ 2 / \ 2 / \ 2 /
----------------------------------------------------------- + -----------------------------------------------------------
/ 2/ pi\\ / 2/ pi\\ / 2/ pi\\ / 2/ pi\\
| cos |3*a - --|| | cos |4*a - --|| | cos |4*a - --|| | cos |5*a - --||
| \ 2 /| | \ 2 /| | \ 2 /| | \ 2 /|
|1 + --------------|*|1 + --------------|*cos(3*a)*cos(4*a) |1 + --------------|*|1 + --------------|*cos(4*a)*cos(5*a)
| 2 | | 2 | | 2 | | 2 |
\ cos (3*a) / \ cos (4*a) / \ cos (4*a) / \ cos (5*a) /
$$\frac{4 \cos{\left(4 a - \frac{\pi}{2} \right)} \cos{\left(5 a - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(4 a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(4 a \right)}}\right) \left(1 + \frac{\cos^{2}{\left(5 a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(5 a \right)}}\right) \cos{\left(4 a \right)} \cos{\left(5 a \right)}} + \frac{4 \cos{\left(3 a - \frac{\pi}{2} \right)} \cos{\left(4 a - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(3 a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(3 a \right)}}\right) \left(1 + \frac{\cos^{2}{\left(4 a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(4 a \right)}}\right) \cos{\left(3 a \right)} \cos{\left(4 a \right)}}$$
// 0 for 6*a mod pi = 0\ // 0 for 8*a mod pi = 0\ // 0 for 8*a mod pi = 0\ // 0 for 10*a mod pi = 0\ || | || | || | || | || 2*tan(3*a) | || 2*tan(4*a) | || 2*tan(4*a) | || 2*tan(5*a) | |<------------- otherwise |*|<------------- otherwise | + |<------------- otherwise |*|<------------- otherwise | || 2 | || 2 | || 2 | || 2 | ||1 + tan (3*a) | ||1 + tan (4*a) | ||1 + tan (4*a) | ||1 + tan (5*a) | \\ / \\ / \\ / \\ /
$$\left(\left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\frac{2 \tan{\left(3 a \right)}}{\tan^{2}{\left(3 a \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 8 a \bmod \pi = 0 \\\frac{2 \tan{\left(4 a \right)}}{\tan^{2}{\left(4 a \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 0 & \text{for}\: 8 a \bmod \pi = 0 \\\frac{2 \tan{\left(4 a \right)}}{\tan^{2}{\left(4 a \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 10 a \bmod \pi = 0 \\\frac{2 \tan{\left(5 a \right)}}{\tan^{2}{\left(5 a \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for 6*a mod pi = 0\ // 0 for 8*a mod pi = 0\ // 0 for 8*a mod pi = 0\ // 0 for 10*a mod pi = 0\ || | || | || | || | || 2*cot(3*a) | || 2*cot(4*a) | || 2*cot(4*a) | || 2*cot(5*a) | |<------------- otherwise |*|<------------- otherwise | + |<------------- otherwise |*|<------------- otherwise | || 2 | || 2 | || 2 | || 2 | ||1 + cot (3*a) | ||1 + cot (4*a) | ||1 + cot (4*a) | ||1 + cot (5*a) | \\ / \\ / \\ / \\ /
$$\left(\left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\frac{2 \cot{\left(3 a \right)}}{\cot^{2}{\left(3 a \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 8 a \bmod \pi = 0 \\\frac{2 \cot{\left(4 a \right)}}{\cot^{2}{\left(4 a \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 0 & \text{for}\: 8 a \bmod \pi = 0 \\\frac{2 \cot{\left(4 a \right)}}{\cot^{2}{\left(4 a \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 10 a \bmod \pi = 0 \\\frac{2 \cot{\left(5 a \right)}}{\cot^{2}{\left(5 a \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
/pi \ /pi \ /pi \ /pi \
4*csc|-- - 4*a|*csc|-- - 3*a| 4*csc|-- - 5*a|*csc|-- - 4*a|
\2 / \2 / \2 / \2 /
----------------------------------------------------------- + -----------------------------------------------------------
/ 2/pi \\ / 2/pi \\ / 2/pi \\ / 2/pi \\
| csc |-- - 3*a|| | csc |-- - 4*a|| | csc |-- - 4*a|| | csc |-- - 5*a||
| \2 /| | \2 /| | \2 /| | \2 /|
|1 + --------------|*|1 + --------------|*csc(3*a)*csc(4*a) |1 + --------------|*|1 + --------------|*csc(4*a)*csc(5*a)
| 2 | | 2 | | 2 | | 2 |
\ csc (3*a) / \ csc (4*a) / \ csc (4*a) / \ csc (5*a) /
$$\frac{4 \csc{\left(- 5 a + \frac{\pi}{2} \right)} \csc{\left(- 4 a + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- 4 a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(4 a \right)}}\right) \left(1 + \frac{\csc^{2}{\left(- 5 a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(5 a \right)}}\right) \csc{\left(4 a \right)} \csc{\left(5 a \right)}} + \frac{4 \csc{\left(- 4 a + \frac{\pi}{2} \right)} \csc{\left(- 3 a + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- 3 a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(3 a \right)}}\right) \left(1 + \frac{\csc^{2}{\left(- 4 a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(4 a \right)}}\right) \csc{\left(3 a \right)} \csc{\left(4 a \right)}}$$
// / 3*pi\ \ // / 3*pi\ \
|| 1 for |8*a + ----| mod 2*pi = 0| || 1 for |8*a + ----| mod 2*pi = 0|
|| \ 2 / | || \ 2 / |
|| | || |
// 0 for 6*a mod pi = 0\ || 2/ pi\ | // 0 for 10*a mod pi = 0\ || 2/ pi\ |
|< |*|<-1 + tan |4*a + --| | + |< |*|<-1 + tan |4*a + --| |
\\sin(6*a) otherwise / || \ 4 / | \\sin(10*a) otherwise / || \ 4 / |
||------------------- otherwise | ||------------------- otherwise |
|| 2/ pi\ | || 2/ pi\ |
|| 1 + tan |4*a + --| | || 1 + tan |4*a + --| |
\\ \ 4 / / \\ \ 4 / /
$$\left(\left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\sin{\left(6 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(8 a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(4 a + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(4 a + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 0 & \text{for}\: 10 a \bmod \pi = 0 \\\sin{\left(10 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(8 a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(4 a + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(4 a + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for 6*a mod pi = 0\ // 0 for 8*a mod pi = 0\ // 0 for 8*a mod pi = 0\ // 0 for 10*a mod pi = 0\
|| | || | || | || |
|| 2 | || 2 | || 2 | || 2 |
||------------------------ otherwise | ||------------------------ otherwise | ||------------------------ otherwise | ||------------------------ otherwise |
|
$$\left(\left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(3 a \right)}}\right) \tan{\left(3 a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 8 a \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(4 a \right)}}\right) \tan{\left(4 a \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 0 & \text{for}\: 8 a \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(4 a \right)}}\right) \tan{\left(4 a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 10 a \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(5 a \right)}}\right) \tan{\left(5 a \right)}} & \text{otherwise} \end{cases}\right)\right)$$
// / 3*pi\ \ // / 3*pi\ \
|| 1 for |8*a + ----| mod 2*pi = 0| || 1 for |8*a + ----| mod 2*pi = 0|
|| \ 2 / | || \ 2 / |
// / 3*pi\ \ || | || | // / 3*pi\ \
|| 1 for |6*a + ----| mod 2*pi = 0| || 2/ pi\ | || 2/ pi\ | || 1 for |10*a + ----| mod 2*pi = 0|
|< \ 2 / |*|<-1 + tan |4*a + --| | + |<-1 + tan |4*a + --| |*|< \ 2 / |
|| | || \ 4 / | || \ 4 / | || |
\\sin(6*a) otherwise / ||------------------- otherwise | ||------------------- otherwise | \\sin(10*a) otherwise /
|| 2/ pi\ | || 2/ pi\ |
|| 1 + tan |4*a + --| | || 1 + tan |4*a + --| |
\\ \ 4 / / \\ \ 4 / /
$$\left(\left(\begin{cases} 1 & \text{for}\: \left(6 a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(6 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(8 a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(4 a + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(4 a + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: \left(8 a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(4 a + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(4 a + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(10 a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(10 a \right)} & \text{otherwise} \end{cases}\right)\right)$$
// / 3*pi\ \ // / 3*pi\ \
|| 1 for |8*a + ----| mod 2*pi = 0| || 1 for |8*a + ----| mod 2*pi = 0|
// 0 for 6*a mod pi = 0\ || \ 2 / | // 0 for 10*a mod pi = 0\ || \ 2 / |
|| | || | || | || |
|| 2*cot(3*a) | || 2/ pi\ | || 2*cot(5*a) | || 2/ pi\ |
|<------------- otherwise |*|<-1 + tan |4*a + --| | + |<------------- otherwise |*|<-1 + tan |4*a + --| |
|| 2 | || \ 4 / | || 2 | || \ 4 / |
||1 + cot (3*a) | ||------------------- otherwise | ||1 + cot (5*a) | ||------------------- otherwise |
\\ / || 2/ pi\ | \\ / || 2/ pi\ |
|| 1 + tan |4*a + --| | || 1 + tan |4*a + --| |
\\ \ 4 / / \\ \ 4 / /
$$\left(\left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\frac{2 \cot{\left(3 a \right)}}{\cot^{2}{\left(3 a \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(8 a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(4 a + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(4 a + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 0 & \text{for}\: 10 a \bmod \pi = 0 \\\frac{2 \cot{\left(5 a \right)}}{\cot^{2}{\left(5 a \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(8 a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(4 a + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(4 a + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for 6*a mod pi = 0\ // 0 for 8*a mod pi = 0\ // 0 for 8*a mod pi = 0\ // 0 for 10*a mod pi = 0\
|| | || | || | || |
|
$$\left(\left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\sin{\left(6 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 8 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 8 a \bmod \pi = 0 \\\sin{\left(8 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 0 & \text{for}\: 8 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 8 a \bmod \pi = 0 \\\sin{\left(8 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 10 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 10 a \bmod \pi = 0 \\\sin{\left(10 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)$$
// / 3*pi\ \ // / 3*pi\ \ // / 3*pi\ \ // / 3*pi\ \ || 1 for |6*a + ----| mod 2*pi = 0| || 1 for |8*a + ----| mod 2*pi = 0| || 1 for |8*a + ----| mod 2*pi = 0| || 1 for |10*a + ----| mod 2*pi = 0| || \ 2 / | || \ 2 / | || \ 2 / | || \ 2 / | || | || | || | || | || 2/ pi\ | || 2/ pi\ | || 2/ pi\ | || 2/ pi\ | |<-1 + tan |3*a + --| |*|<-1 + tan |4*a + --| | + |<-1 + tan |4*a + --| |*|<-1 + tan |5*a + --| | || \ 4 / | || \ 4 / | || \ 4 / | || \ 4 / | ||------------------- otherwise | ||------------------- otherwise | ||------------------- otherwise | ||------------------- otherwise | || 2/ pi\ | || 2/ pi\ | || 2/ pi\ | || 2/ pi\ | || 1 + tan |3*a + --| | || 1 + tan |4*a + --| | || 1 + tan |4*a + --| | || 1 + tan |5*a + --| | \\ \ 4 / / \\ \ 4 / / \\ \ 4 / / \\ \ 4 / /
$$\left(\left(\begin{cases} 1 & \text{for}\: \left(6 a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(3 a + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(3 a + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(8 a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(4 a + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(4 a + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: \left(8 a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(4 a + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(4 a + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(10 a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(5 a + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(5 a + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for 6*a mod pi = 0\ // 0 for 8*a mod pi = 0\ // 0 for 8*a mod pi = 0\ // 0 for 10*a mod pi = 0\
|| | || | || | || |
|| sin(6*a) | || sin(8*a) | || sin(8*a) | || sin(10*a) |
||--------------------------- otherwise | ||--------------------------- otherwise | ||--------------------------- otherwise | ||--------------------------- otherwise |
|
$$\left(\left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\frac{\sin{\left(6 a \right)}}{\left(1 + \frac{\sin^{2}{\left(6 a \right)}}{4 \sin^{4}{\left(3 a \right)}}\right) \sin^{2}{\left(3 a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 8 a \bmod \pi = 0 \\\frac{\sin{\left(8 a \right)}}{\left(1 + \frac{\sin^{2}{\left(8 a \right)}}{4 \sin^{4}{\left(4 a \right)}}\right) \sin^{2}{\left(4 a \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 0 & \text{for}\: 8 a \bmod \pi = 0 \\\frac{\sin{\left(8 a \right)}}{\left(1 + \frac{\sin^{2}{\left(8 a \right)}}{4 \sin^{4}{\left(4 a \right)}}\right) \sin^{2}{\left(4 a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 10 a \bmod \pi = 0 \\\frac{\sin{\left(10 a \right)}}{\left(1 + \frac{\sin^{2}{\left(10 a \right)}}{4 \sin^{4}{\left(5 a \right)}}\right) \sin^{2}{\left(5 a \right)}} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 3*a mod 2*pi = 0\ // 1 for 5*a mod 2*pi = 0\
// 1 for 2*a mod pi = 0\ || | // 1 for 2*a mod pi = 0\ || |
|| | || 2 | || | || 2 |
|| 2 | ||/ 2/3*a\\ | || 2 | ||/ 2/5*a\\ |
||/ 2 \ | |||-1 + cot |---|| | ||/ 2 \ | |||-1 + cot |---|| |
4*|<\-1 + cot (2*a)/ |*|<\ \ 2 // | 4*|<\-1 + cot (2*a)/ |*|<\ \ 2 // |
||----------------- otherwise | ||----------------- otherwise | ||----------------- otherwise | ||----------------- otherwise |
|| 2 | || 2 | || 2 | || 2 |
|| / 2 \ | || / 2/3*a\\ | || / 2 \ | || / 2/5*a\\ |
\\ \1 + cot (2*a)/ / || |1 + cot |---|| | \\ \1 + cot (2*a)/ / || |1 + cot |---|| |
\\ \ \ 2 // / \\ \ \ 2 // /
------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------
cot(3*a)*cot(4*a) cot(4*a)*cot(5*a)
$$\left(\frac{4 \left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{\left(\cot^{2}{\left(2 a \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(2 a \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{3 a}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{3 a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)}{\cot{\left(3 a \right)} \cot{\left(4 a \right)}}\right) + \left(\frac{4 \left(\begin{cases} 1 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{\left(\cot^{2}{\left(2 a \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(2 a \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 5 a \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{5 a}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{5 a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)}{\cot{\left(4 a \right)} \cot{\left(5 a \right)}}\right)$$
// 0 for 6*a mod pi = 0\ // 0 for 8*a mod pi = 0\ // 0 for 8*a mod pi = 0\ // 0 for 10*a mod pi = 0\ || | || | || | || | ||/ 0 for 6*a mod pi = 0 | ||/ 0 for 8*a mod pi = 0 | ||/ 0 for 8*a mod pi = 0 | ||/ 0 for 10*a mod pi = 0 | ||| | ||| | ||| | ||| | |<| 2*cot(3*a) |*|<| 2*cot(4*a) | + |<| 2*cot(4*a) |*|<| 2*cot(5*a) | ||<------------- otherwise otherwise | ||<------------- otherwise otherwise | ||<------------- otherwise otherwise | ||<------------- otherwise otherwise | ||| 2 | ||| 2 | ||| 2 | ||| 2 | |||1 + cot (3*a) | |||1 + cot (4*a) | |||1 + cot (4*a) | |||1 + cot (5*a) | \\\ / \\\ / \\\ / \\\ /
$$\left(\left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\frac{2 \cot{\left(3 a \right)}}{\cot^{2}{\left(3 a \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 8 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 8 a \bmod \pi = 0 \\\frac{2 \cot{\left(4 a \right)}}{\cot^{2}{\left(4 a \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 0 & \text{for}\: 8 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 8 a \bmod \pi = 0 \\\frac{2 \cot{\left(4 a \right)}}{\cot^{2}{\left(4 a \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 10 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 10 a \bmod \pi = 0 \\\frac{2 \cot{\left(5 a \right)}}{\cot^{2}{\left(5 a \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for 6*a mod pi = 0\ // 0 for 8*a mod pi = 0\ // 0 for 8*a mod pi = 0\ // 0 for 10*a mod pi = 0\
|| | || | || | || |
|| / pi\ | || / pi\ | || / pi\ | || / pi\ |
|| 2*sec|3*a - --| | || 2*sec|4*a - --| | || 2*sec|4*a - --| | || 2*sec|5*a - --| |
|| \ 2 / | || \ 2 / | || \ 2 / | || \ 2 / |
||----------------------------- otherwise | ||----------------------------- otherwise | ||----------------------------- otherwise | ||----------------------------- otherwise |
|
$$\left(\left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\frac{2 \sec{\left(3 a - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(3 a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(3 a \right)}}\right) \sec{\left(3 a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 8 a \bmod \pi = 0 \\\frac{2 \sec{\left(4 a - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(4 a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(4 a \right)}}\right) \sec{\left(4 a \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 0 & \text{for}\: 8 a \bmod \pi = 0 \\\frac{2 \sec{\left(4 a - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(4 a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(4 a \right)}}\right) \sec{\left(4 a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 10 a \bmod \pi = 0 \\\frac{2 \sec{\left(5 a - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(5 a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(5 a \right)}}\right) \sec{\left(5 a \right)}} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for 6*a mod pi = 0\ // 0 for 8*a mod pi = 0\ // 0 for 8*a mod pi = 0\ // 0 for 10*a mod pi = 0\ || | || | || | || | || 2*cos(3*a) | || 2*cos(4*a) | || 2*cos(4*a) | || 2*cos(5*a) | ||---------------------------------- otherwise | ||---------------------------------- otherwise | ||---------------------------------- otherwise | ||---------------------------------- otherwise | ||/ 2 \ | ||/ 2 \ | ||/ 2 \ | ||/ 2 \ | |<| cos (3*a) | / pi\ |*|<| cos (4*a) | / pi\ | + |<| cos (4*a) | / pi\ |*|<| cos (5*a) | / pi\ | |||1 + --------------|*cos|3*a - --| | |||1 + --------------|*cos|4*a - --| | |||1 + --------------|*cos|4*a - --| | |||1 + --------------|*cos|5*a - --| | ||| 2/ pi\| \ 2 / | ||| 2/ pi\| \ 2 / | ||| 2/ pi\| \ 2 / | ||| 2/ pi\| \ 2 / | ||| cos |3*a - --|| | ||| cos |4*a - --|| | ||| cos |4*a - --|| | ||| cos |5*a - --|| | ||\ \ 2 // | ||\ \ 2 // | ||\ \ 2 // | ||\ \ 2 // | \\ / \\ / \\ / \\ /
$$\left(\left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\frac{2 \cos{\left(3 a \right)}}{\left(\frac{\cos^{2}{\left(3 a \right)}}{\cos^{2}{\left(3 a - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(3 a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 8 a \bmod \pi = 0 \\\frac{2 \cos{\left(4 a \right)}}{\left(\frac{\cos^{2}{\left(4 a \right)}}{\cos^{2}{\left(4 a - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(4 a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 0 & \text{for}\: 8 a \bmod \pi = 0 \\\frac{2 \cos{\left(4 a \right)}}{\left(\frac{\cos^{2}{\left(4 a \right)}}{\cos^{2}{\left(4 a - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(4 a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 10 a \bmod \pi = 0 \\\frac{2 \cos{\left(5 a \right)}}{\left(\frac{\cos^{2}{\left(5 a \right)}}{\cos^{2}{\left(5 a - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(5 a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for 6*a mod pi = 0\ // 0 for 8*a mod pi = 0\ // 0 for 8*a mod pi = 0\ // 0 for 10*a mod pi = 0\ || | || | || | || | || 2*csc(3*a) | || 2*csc(4*a) | || 2*csc(4*a) | || 2*csc(5*a) | ||---------------------------------- otherwise | ||---------------------------------- otherwise | ||---------------------------------- otherwise | ||---------------------------------- otherwise | ||/ 2 \ | ||/ 2 \ | ||/ 2 \ | ||/ 2 \ | |<| csc (3*a) | /pi \ |*|<| csc (4*a) | /pi \ | + |<| csc (4*a) | /pi \ |*|<| csc (5*a) | /pi \ | |||1 + --------------|*csc|-- - 3*a| | |||1 + --------------|*csc|-- - 4*a| | |||1 + --------------|*csc|-- - 4*a| | |||1 + --------------|*csc|-- - 5*a| | ||| 2/pi \| \2 / | ||| 2/pi \| \2 / | ||| 2/pi \| \2 / | ||| 2/pi \| \2 / | ||| csc |-- - 3*a|| | ||| csc |-- - 4*a|| | ||| csc |-- - 4*a|| | ||| csc |-- - 5*a|| | ||\ \2 // | ||\ \2 // | ||\ \2 // | ||\ \2 // | \\ / \\ / \\ / \\ /
$$\left(\left(\begin{cases} 0 & \text{for}\: 6 a \bmod \pi = 0 \\\frac{2 \csc{\left(3 a \right)}}{\left(\frac{\csc^{2}{\left(3 a \right)}}{\csc^{2}{\left(- 3 a + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- 3 a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 8 a \bmod \pi = 0 \\\frac{2 \csc{\left(4 a \right)}}{\left(\frac{\csc^{2}{\left(4 a \right)}}{\csc^{2}{\left(- 4 a + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- 4 a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 0 & \text{for}\: 8 a \bmod \pi = 0 \\\frac{2 \csc{\left(4 a \right)}}{\left(\frac{\csc^{2}{\left(4 a \right)}}{\csc^{2}{\left(- 4 a + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- 4 a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 10 a \bmod \pi = 0 \\\frac{2 \csc{\left(5 a \right)}}{\left(\frac{\csc^{2}{\left(5 a \right)}}{\csc^{2}{\left(- 5 a + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- 5 a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right)$$
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))*Piecewise((0, Mod(8*a = pi, 0)), (2*csc(4*a)/((1 + csc(4*a)^2/csc(pi/2 - 4*a)^2)*csc(pi/2 - 4*a)), True)) + Piecewise((0, Mod(8*a = pi, 0)), (2*csc(4*a)/((1 + csc(4*a)^2/csc(pi/2 - 4*a)^2)*csc(pi/2 - 4*a)), True))*Piecewise((0, Mod(10*a = pi, 0)), (2*csc(5*a)/((1 + csc(5*a)^2/csc(pi/2 - 5*a)^2)*csc(pi/2 - 5*a)), True))