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