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