Господин Экзамен
cos(2*a)+2*sin(pi-a)^2 если a=4
Решение
Вы ввели
[src]
2 cos(2*a) + 2*sin (pi - a)
$$2 \sin^{2}{\left(- a + \pi \right)} + \cos{\left(2 a \right)}$$
cos(2*a) + 2*sin(pi - a)^2
Объединение рациональных выражений
[src]
2 2*sin (a) + cos(2*a)
$$2 \sin^{2}{\left(a \right)} + \cos{\left(2 a \right)}$$
2*sin(a)^2 + cos(2*a)
Общий знаменатель
[src]
2 2*sin (a) + cos(2*a)
$$2 \sin^{2}{\left(a \right)} + \cos{\left(2 a \right)}$$
2*sin(a)^2 + cos(2*a)
Степени
[src]
2 2*sin (a) + cos(2*a)
$$2 \sin^{2}{\left(a \right)} + \cos{\left(2 a \right)}$$
2 -2*I*a 2*I*a / I*(a - pi) I*(pi - a)\ e e \- e + e / ------- + ------ - ------------------------------ 2 2 2
$$- \frac{\left(e^{i \left(- a + \pi\right)} - e^{i \left(a - \pi\right)}\right)^{2}}{2} + \frac{e^{2 i a}}{2} + \frac{e^{- 2 i a}}{2}$$
exp(-2*i*a)/2 + exp(2*i*a)/2 - (-exp(i*(a - pi)) + exp(i*(pi - a)))^2/2
Комбинаторика
[src]
2 2*sin (a) + cos(2*a)
$$2 \sin^{2}{\left(a \right)} + \cos{\left(2 a \right)}$$
2*sin(a)^2 + cos(2*a)
Рациональный знаменатель
[src]
2 2*sin (a) + cos(2*a)
$$2 \sin^{2}{\left(a \right)} + \cos{\left(2 a \right)}$$
2*sin(a)^2 + cos(2*a)
Тригонометрическая часть
[src]
1
$$1$$
2 2*sin (a) + cos(2*a)
$$2 \sin^{2}{\left(a \right)} + \cos{\left(2 a \right)}$$
2 2 -1 + 2*cos (a) + 2*sin (a)
$$2 \sin^{2}{\left(a \right)} + 2 \cos^{2}{\left(a \right)} - 1$$
2 /pi \
2*sin (a) + sin|-- + 2*a|
\2 /
$$2 \sin^{2}{\left(a \right)} + \sin{\left(2 a + \frac{\pi}{2} \right)}$$
2/ pi\
2*cos |a - --| + cos(2*a)
\ 2 /
$$2 \cos^{2}{\left(a - \frac{\pi}{2} \right)} + \cos{\left(2 a \right)}$$
1 2
-------- + -------
sec(2*a) 2
csc (a)
$$\frac{1}{\sec{\left(2 a \right)}} + \frac{2}{\csc^{2}{\left(a \right)}}$$
1 2
-------- + ------------
sec(2*a) 2/ pi\
sec |a - --|
\ 2 /
$$\frac{1}{\sec{\left(2 a \right)}} + \frac{2}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}$$
1 2 ------------- + ------- /pi \ 2 csc|-- - 2*a| csc (a) \2 /
$$\frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}} + \frac{2}{\csc^{2}{\left(a \right)}}$$
1 2
-------- + ------------
sec(2*a) 2/pi \
sec |-- - a|
\2 /
$$\frac{1}{\sec{\left(2 a \right)}} + \frac{2}{\sec^{2}{\left(- a + \frac{\pi}{2} \right)}}$$
1 2 ------------- + ------------ /pi \ 2 csc|-- - 2*a| csc (pi - a) \2 /
$$\frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}} + \frac{2}{\csc^{2}{\left(- a + \pi \right)}}$$
2 4 - 4*cos(a) - 2*(1 - cos(a)) + cos(2*a)
$$- 2 \left(- \cos{\left(a \right)} + 1\right)^{2} - 4 \cos{\left(a \right)} + \cos{\left(2 a \right)} + 4$$
2
/ 2/a pi\\ 2
|1 - cot |- + --|| *(1 + sin(a))
\ \2 4 //
--------------------------------- + cos(2*a)
2
$$\frac{\left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\sin{\left(a \right)} + 1\right)^{2}}{2} + \cos{\left(2 a \right)}$$
2/a\
2 8*tan |-|
1 - tan (a) \2/
----------- + --------------
2 2
1 + tan (a) / 2/a\\
|1 + tan |-||
\ \2//
$$\frac{- \tan^{2}{\left(a \right)} + 1}{\tan^{2}{\left(a \right)} + 1} + \frac{8 \tan^{2}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}$$
2 4/a\
32*sin (a)*sin |-|
\2/
---------------------- + cos(2*a)
2
/ 2 4/a\\
|sin (a) + 4*sin |-||
\ \2//
$$\frac{32 \sin^{4}{\left(\frac{a}{2} \right)} \sin^{2}{\left(a \right)}}{\left(4 \sin^{4}{\left(\frac{a}{2} \right)} + \sin^{2}{\left(a \right)}\right)^{2}} + \cos{\left(2 a \right)}$$
/ pi\ 2/a\
2*tan|a + --| 8*tan |-|
\ 4 / \2/
---------------- + --------------
2/ pi\ 2
1 + tan |a + --| / 2/a\\
\ 4 / |1 + tan |-||
\ \2//
$$\frac{2 \tan{\left(a + \frac{\pi}{4} \right)}}{\tan^{2}{\left(a + \frac{\pi}{4} \right)} + 1} + \frac{8 \tan^{2}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}$$
/ pi\ 2/a\
2*tan|a + --| 8*cot |-|
\ 4 / \2/
---------------- + --------------
2/ pi\ 2
1 + tan |a + --| / 2/a\\
\ 4 / |1 + cot |-||
\ \2//
$$\frac{8 \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} + \frac{2 \tan{\left(a + \frac{\pi}{4} \right)}}{\tan^{2}{\left(a + \frac{\pi}{4} \right)} + 1}$$
1
1 - -------
2
cot (a) 8
----------- + ----------------------
1 2
1 + ------- / 1 \ 2/a\
2 |1 + -------| *cot |-|
cot (a) | 2/a\| \2/
| cot |-||
\ \2//
$$\frac{1 - \frac{1}{\cot^{2}{\left(a \right)}}}{1 + \frac{1}{\cot^{2}{\left(a \right)}}} + \frac{8}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right)^{2} \cot^{2}{\left(\frac{a}{2} \right)}}$$
2
/ 2/a pi\\
2 2*|-1 + tan |- + --||
-1 + cot (a) \ \2 4 //
------------ + ----------------------
2 2
1 + cot (a) / 2/a pi\\
|1 + tan |- + --||
\ \2 4 //
$$\frac{2 \left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} + \frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1}$$
2
/ 2/a pi\\
2 2*|1 - cot |- + --||
1 - tan (a) \ \2 4 //
----------- + ---------------------
2 2
1 + tan (a) / 2/a pi\\
|1 + cot |- + --||
\ \2 4 //
$$\frac{- \tan^{2}{\left(a \right)} + 1}{\tan^{2}{\left(a \right)} + 1} + \frac{2 \left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
// 0 for a mod pi = 0\ || | // 1 for a mod pi = 0\ 2*|< 2 | + |< | ||sin (a) otherwise | \\cos(2*a) otherwise / \\ /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\ // 1 for a mod pi = 0\ || | || | 2*|< 2 | + |< /pi \ | ||sin (a) otherwise | ||sin|-- + 2*a| otherwise | \\ / \\ \2 / /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(2 a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\ || | // 1 for a mod pi = 0\ 2*|< 2/ pi\ | + |< | ||cos |a - --| otherwise | \\cos(2*a) otherwise / \\ \ 2 / /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\cos^{2}{\left(a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\ // /pi \ \ || | || 0 for |-- + 2*a| mod pi = 0| 2*|< 2 | + |< \2 / | ||sin (a) otherwise | || | \\ / \\cos(2*a) otherwise /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \left(2 a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\ || | // 1 for a mod pi = 0\ || 1 | || | 2*|<------------ otherwise | + |< 1 | || 2/ pi\ | ||-------- otherwise | ||sec |a - --| | \\sec(2*a) / \\ \ 2 / /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sec{\left(2 a \right)}} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\ // 1 for a mod pi = 0\ || | || | || 1 | || 1 | 2*|<------- otherwise | + |<------------- otherwise | || 2 | || /pi \ | ||csc (a) | ||csc|-- - 2*a| | \\ / \\ \2 / /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\csc^{2}{\left(a \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
4
4*sin (a)
1 - --------- 4/a\
2 32*sin |-|
sin (2*a) \2/
------------- + ------------------------
4 2
4*sin (a) / 4/a\\
1 + --------- | 4*sin |-||
2 | \2/| 2
sin (2*a) |1 + ---------| *sin (a)
| 2 |
\ sin (a) /
$$\frac{- \frac{4 \sin^{4}{\left(a \right)}}{\sin^{2}{\left(2 a \right)}} + 1}{\frac{4 \sin^{4}{\left(a \right)}}{\sin^{2}{\left(2 a \right)}} + 1} + \frac{32 \sin^{4}{\left(\frac{a}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right)^{2} \sin^{2}{\left(a \right)}}$$
// / 3*pi\ \ || 1 for |a + ----| mod 2*pi = 0| || \ 2 / | // 1 for a mod pi = 0\ 2*|< | + |< | || 4/a\ 2/a\ | \\cos(2*a) otherwise / ||- 4*cos |-| + 4*cos |-| otherwise | \\ \2/ \2/ /
$$\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\- 4 \cos^{4}{\left(\frac{a}{2} \right)} + 4 \cos^{2}{\left(\frac{a}{2} \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for a mod pi = 0\ || | || 2/a\ | // 1 for a mod pi = 0\ || 4*cot |-| | || | || \2/ | || 2 | 2*|<-------------- otherwise | + |<-1 + cot (a) | || 2 | ||------------ otherwise | ||/ 2/a\\ | || 2 | |||1 + cot |-|| | \\1 + cot (a) / ||\ \2// | \\ /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\ || | || 2/a\ | // 1 for a mod pi = 0\ || 4*tan |-| | || | || \2/ | || 2 | 2*|<-------------- otherwise | + |<1 - tan (a) | || 2 | ||----------- otherwise | ||/ 2/a\\ | || 2 | |||1 + tan |-|| | \\1 + tan (a) / ||\ \2// | \\ /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{4 \tan^{2}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{- \tan^{2}{\left(a \right)} + 1}{\tan^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right)$$
2/ pi\
cos |a - --|
\ 2 /
1 - ------------ 2/a pi\
2 8*cos |- - --|
cos (a) \2 2 /
---------------- + ---------------------------
2/ pi\ 2
cos |a - --| / 2/a pi\\
\ 2 / | cos |- - --||
1 + ------------ | \2 2 /| 2/a\
2 |1 + ------------| *cos |-|
cos (a) | 2/a\ | \2/
| cos |-| |
\ \2/ /
$$\frac{1 - \frac{\cos^{2}{\left(a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(a \right)}}}{1 + \frac{\cos^{2}{\left(a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(a \right)}}} + \frac{8 \cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right)^{2} \cos^{2}{\left(\frac{a}{2} \right)}}$$
2
sec (a)
1 - ------------
2/ pi\ 2/a\
sec |a - --| 8*sec |-|
\ 2 / \2/
---------------- + --------------------------------
2 2
sec (a) / 2/a\ \
1 + ------------ | sec |-| |
2/ pi\ | \2/ | 2/a pi\
sec |a - --| |1 + ------------| *sec |- - --|
\ 2 / | 2/a pi\| \2 2 /
| sec |- - --||
\ \2 2 //
$$\frac{- \frac{\sec^{2}{\left(a \right)}}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} + 1}{\frac{\sec^{2}{\left(a \right)}}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} + 1} + \frac{8 \sec^{2}{\left(\frac{a}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}$$
// 0 for a mod pi = 0\
|| |
|| 2 |
|| sin (a) |
||------------------------ otherwise |
|| 2 | // 1 for a mod pi = 0\
2*|
$$\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{\sin^{2}{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right)$$
2/pi \
csc |-- - a|
\2 /
1 - ------------ 2/pi a\
2 8*csc |-- - -|
csc (a) \2 2/
---------------- + ---------------------------
2/pi \ 2
csc |-- - a| / 2/pi a\\
\2 / | csc |-- - -||
1 + ------------ | \2 2/| 2/a\
2 |1 + ------------| *csc |-|
csc (a) | 2/a\ | \2/
| csc |-| |
\ \2/ /
$$\frac{1 - \frac{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(a \right)}}}{1 + \frac{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(a \right)}}} + \frac{8 \csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right)^{2} \csc^{2}{\left(\frac{a}{2} \right)}}$$
// 0 for a mod pi = 0\ // 1 for a mod pi = 0\
|| | || |
|| 4 | || 1 |
||---------------------- otherwise | ||-1 + ------- |
|| 2 | || 2 |
2*|
$$\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{4}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right)^{2} \tan^{2}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(a \right)}}}{1 + \frac{1}{\tan^{2}{\left(a \right)}}} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\
|| | // 1 for a mod pi = 0\
||/ 0 for a mod pi = 0 | || |
2*|<| | + |
$$\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin^{2}{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\ // /pi \ \ || | || 0 for |-- + 2*a| mod pi = 0| || 2/a\ | || \2 / | || 4*cot |-| | || | || \2/ | || / pi\ | 2*|<-------------- otherwise | + |< 2*cot|a + --| | || 2 | || \ 4 / | ||/ 2/a\\ | ||---------------- otherwise | |||1 + cot |-|| | || 2/ pi\ | ||\ \2// | ||1 + cot |a + --| | \\ / \\ \ 4 / /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \left(2 a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(a + \frac{\pi}{4} \right)}}{\cot^{2}{\left(a + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// / 3*pi\ \ || 1 for |a + ----| mod 2*pi = 0| || \ 2 / | || | // 1 for a mod pi = 0\ || 2 | || | ||/ 2/a pi\\ | || 2 | 2*|<|-1 + tan |- + --|| | + |<-1 + cot (a) | ||\ \2 4 // | ||------------ otherwise | ||-------------------- otherwise | || 2 | || 2 | \\1 + cot (a) / ||/ 2/a pi\\ | |||1 + tan |- + --|| | \\\ \2 4 // /
$$\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for a mod pi = 0\ // 1 for a mod pi = 0\
|| | || |
|| 2 | || 2 |
|| sin (a) | || sin (2*a) |
||------------------------ otherwise | ||-1 + --------- |
|| 2 | || 4 |
2*|
$$\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{\sin^{2}{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(2 a \right)}}{4 \sin^{4}{\left(a \right)}}}{1 + \frac{\sin^{2}{\left(2 a \right)}}{4 \sin^{4}{\left(a \right)}}} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\ || | ||/ 0 for a mod pi = 0 | // 1 for a mod pi = 0\ ||| | || | ||| 2/a\ | ||/ 1 for a mod pi = 0 | ||| 4*cot |-| | ||| | 2*|<| \2/ | + |<| 2 | ||<-------------- otherwise otherwise | ||<-1 + cot (a) otherwise | ||| 2 | |||------------ otherwise | |||/ 2/a\\ | ||| 2 | ||||1 + cot |-|| | \\\1 + cot (a) / |||\ \2// | \\\ /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\ // 1 for a mod pi = 0\
|| | || |
|| 2/a\ | || 2 |
|| 4*cos |-| | || cos (a) |
|| \2/ | ||-1 + ------------ |
||-------------------------------- otherwise | || 2/ pi\ |
|| 2 | || cos |a - --| |
2*|
$$\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{4 \cos^{2}{\left(\frac{a}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\frac{\cos^{2}{\left(a \right)}}{\cos^{2}{\left(a - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(a \right)}}{\cos^{2}{\left(a - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\ // 1 for a mod pi = 0\
|| | || |
|| 2/a pi\ | || 2/ pi\ |
|| 4*sec |- - --| | || sec |a - --| |
|| \2 2 / | || \ 2 / |
||--------------------------- otherwise | ||-1 + ------------ |
|| 2 | || 2 |
2*|
$$\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{4 \sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right)^{2} \sec^{2}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(a \right)}}}{1 + \frac{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(a \right)}}} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\ // 1 for a mod pi = 0\
|| | || |
|| 2/a\ | || 2 |
|| 4*csc |-| | || csc (a) |
|| \2/ | ||-1 + ------------ |
||-------------------------------- otherwise | || 2/pi \ |
|| 2 | || csc |-- - a| |
2*|
$$\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{4 \csc^{2}{\left(\frac{a}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2} \csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\frac{\csc^{2}{\left(a \right)}}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(a \right)}}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)$$
2*Piecewise((0, Mod(a = pi, 0)), (4*csc(a/2)^2/((1 + csc(a/2)^2/csc(pi/2 - a/2)^2)^2*csc(pi/2 - a/2)^2), True)) + Piecewise((1, Mod(a = pi, 0)), ((-1 + csc(a)^2/csc(pi/2 - a)^2)/(1 + csc(a)^2/csc(pi/2 - a)^2), True))