Господин Экзамен
cos(t)^2-sin(t)^2/(tan(-t)*cot(t)) если t=-3/2
Решение
Вы ввели
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2
2 sin (t)
cos (t) - --------------
tan(-t)*cot(t)
$$\cos^{2}{\left(t \right)} - \frac{\sin^{2}{\left(t \right)}}{\tan{\left(- t \right)} \cot{\left(t \right)}}$$
cos(t)^2 - sin(t)^2/(tan(-t)*cot(t))
Рациональный знаменатель
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2
2 sin (t)
cos (t) + -------------
cot(t)*tan(t)
$$\cos^{2}{\left(t \right)} + \frac{\sin^{2}{\left(t \right)}}{\tan{\left(t \right)} \cot{\left(t \right)}}$$
2 2
sin (t) + cos (t)*cot(t)*tan(t)
-------------------------------
cot(t)*tan(t)
$$\frac{\cos^{2}{\left(t \right)} \tan{\left(t \right)} \cot{\left(t \right)} + \sin^{2}{\left(t \right)}}{\tan{\left(t \right)} \cot{\left(t \right)}}$$
(sin(t)^2 + cos(t)^2*cot(t)*tan(t))/(cot(t)*tan(t))
Объединение рациональных выражений
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2 2
sin (t) + cos (t)*cot(t)*tan(t)
-------------------------------
cot(t)*tan(t)
$$\frac{\cos^{2}{\left(t \right)} \tan{\left(t \right)} \cot{\left(t \right)} + \sin^{2}{\left(t \right)}}{\tan{\left(t \right)} \cot{\left(t \right)}}$$
(sin(t)^2 + cos(t)^2*cot(t)*tan(t))/(cot(t)*tan(t))
Комбинаторика
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2 2
sin (t) + cos (t)*cot(t)*tan(t)
-------------------------------
cot(t)*tan(t)
$$\frac{\cos^{2}{\left(t \right)} \tan{\left(t \right)} \cot{\left(t \right)} + \sin^{2}{\left(t \right)}}{\tan{\left(t \right)} \cot{\left(t \right)}}$$
(sin(t)^2 + cos(t)^2*cot(t)*tan(t))/(cot(t)*tan(t))
Степени
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2
2 sin (t)
cos (t) + -------------
cot(t)*tan(t)
$$\cos^{2}{\left(t \right)} + \frac{\sin^{2}{\left(t \right)}}{\tan{\left(t \right)} \cot{\left(t \right)}}$$
2 / I*t -I*t\ / -I*t I*t\ / I*t -I*t\ |e e | I*\- e + e /*\e + e / |---- + -----| - --------------------------------- \ 2 2 / 4*cot(t)
$$\left(\frac{e^{i t}}{2} + \frac{e^{- i t}}{2}\right)^{2} - \frac{i \left(e^{i t} - e^{- i t}\right) \left(e^{i t} + e^{- i t}\right)}{4 \cot{\left(t \right)}}$$
(exp(i*t)/2 + exp(-i*t)/2)^2 - i*(-exp(-i*t) + exp(i*t))*(exp(i*t) + exp(-i*t))/(4*cot(t))
Тригонометрическая часть
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1
$$1$$
2 2 cos (t) + sin (t)
$$\sin^{2}{\left(t \right)} + \cos^{2}{\left(t \right)}$$
2 2/ pi\
sin (t) + sin |t + --|
\ 2 /
$$\sin^{2}{\left(t \right)} + \sin^{2}{\left(t + \frac{\pi}{2} \right)}$$
2 2/ pi\
cos (t) + cos |t - --|
\ 2 /
$$\cos^{2}{\left(t \right)} + \cos^{2}{\left(t - \frac{\pi}{2} \right)}$$
1 1 ------- + ------- 2 2 csc (t) sec (t)
$$\frac{1}{\sec^{2}{\left(t \right)}} + \frac{1}{\csc^{2}{\left(t \right)}}$$
1 1
------- + ------------
2 2/ pi\
sec (t) sec |t - --|
\ 2 /
$$\frac{1}{\sec^{2}{\left(t - \frac{\pi}{2} \right)}} + \frac{1}{\sec^{2}{\left(t \right)}}$$
1 1
------- + ------------
2 2/pi \
csc (t) csc |-- - t|
\2 /
$$\frac{1}{\csc^{2}{\left(- t + \frac{\pi}{2} \right)}} + \frac{1}{\csc^{2}{\left(t \right)}}$$
1 1
------- + ------------
2 2/pi \
sec (t) sec |-- - t|
\2 /
$$\frac{1}{\sec^{2}{\left(- t + \frac{\pi}{2} \right)}} + \frac{1}{\sec^{2}{\left(t \right)}}$$
1 1
------------ + ------------
2 2/pi \
csc (pi - t) csc |-- - t|
\2 /
$$\frac{1}{\csc^{2}{\left(- t + \frac{\pi}{2} \right)}} + \frac{1}{\csc^{2}{\left(- t + \pi \right)}}$$
2
2 sin (t)
cos (t) + -------------
cot(t)*tan(t)
$$\cos^{2}{\left(t \right)} + \frac{\sin^{2}{\left(t \right)}}{\tan{\left(t \right)} \cot{\left(t \right)}}$$
5 cos(2*t) 2 - + -------- - (1 - cos(t)) - 2*cos(t) 2 2
$$- \left(- \cos{\left(t \right)} + 1\right)^{2} - 2 \cos{\left(t \right)} + \frac{\cos{\left(2 t \right)}}{2} + \frac{5}{2}$$
1 cos(2*t)
- - --------
1 cos(2*t) 2 2
- + -------- + -------------
2 2 cot(t)*tan(t)
$$\frac{\cos{\left(2 t \right)}}{2} + \frac{1}{2} + \frac{- \frac{\cos{\left(2 t \right)}}{2} + \frac{1}{2}}{\tan{\left(t \right)} \cot{\left(t \right)}}$$
2
/ 2/t pi\\ 2
|1 - cot |- + --|| *(1 + sin(t))
1 + cos(2*t) \ \2 4 //
------------ + ---------------------------------
2 4
$$\frac{\left(- \cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\sin{\left(t \right)} + 1\right)^{2}}{4} + \frac{\cos{\left(2 t \right)} + 1}{2}$$
2/t pi\
4*tan |- + --|
1 - cos(2*t) \2 4 /
------------ + -------------------
2 2
/ 2/t pi\\
|1 + tan |- + --||
\ \2 4 //
$$\frac{- \cos{\left(2 t \right)} + 1}{2} + \frac{4 \tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
2
/ 2/t\\ 2/t\
|1 - tan |-|| 4*tan |-|
\ \2// \2/
-------------- + --------------
2 2
/ 2/t\\ / 2/t\\
|1 + tan |-|| |1 + tan |-||
\ \2// \ \2//
$$\frac{\left(- \tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} + \frac{4 \tan^{2}{\left(\frac{t}{2} \right)}}{\left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}}$$
2 2
1 sin (t) cos (t)
2 2 - + ------- - -------
1 cos (t) sin (t) 2 2 2
- + ------- - ------- + ---------------------
2 2 2 cot(t)*tan(t)
$$- \frac{\sin^{2}{\left(t \right)}}{2} + \frac{\cos^{2}{\left(t \right)}}{2} + \frac{1}{2} + \frac{\frac{\sin^{2}{\left(t \right)}}{2} - \frac{\cos^{2}{\left(t \right)}}{2} + \frac{1}{2}}{\tan{\left(t \right)} \cot{\left(t \right)}}$$
2/t\ 2/t pi\
4*cot |-| 4*tan |- + --|
\2/ \2 4 /
-------------- + -------------------
2 2
/ 2/t\\ / 2/t pi\\
|1 + cot |-|| |1 + tan |- + --||
\ \2// \ \2 4 //
$$\frac{4 \cot^{2}{\left(\frac{t}{2} \right)}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} + \frac{4 \tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
2/t\ 2/t pi\
4*tan |-| 4*tan |- + --|
\2/ \2 4 /
-------------- + -------------------
2 2
/ 2/t\\ / 2/t pi\\
|1 + tan |-|| |1 + tan |- + --||
\ \2// \ \2 4 //
$$\frac{4 \tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} + \frac{4 \tan^{2}{\left(\frac{t}{2} \right)}}{\left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}}$$
// 0 for t mod pi = 0\ // 1 for t mod 2*pi = 0\ || | || | |< 2 | + |< 2 | ||sin (t) otherwise | ||cos (t) otherwise | \\ / \\ /
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\sin^{2}{\left(t \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\cos^{2}{\left(t \right)} & \text{otherwise} \end{cases}\right)$$
2
/ 1 \
|1 - -------|
| 2/t\|
| cot |-||
\ \2// 4
-------------- + ----------------------
2 2
/ 1 \ / 1 \ 2/t\
|1 + -------| |1 + -------| *cot |-|
| 2/t\| | 2/t\| \2/
| cot |-|| | cot |-||
\ \2// \ \2//
$$\frac{\left(1 - \frac{1}{\cot^{2}{\left(\frac{t}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{t}{2} \right)}}\right)^{2}} + \frac{4}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \cot^{2}{\left(\frac{t}{2} \right)}}$$
// 0 for t mod pi = 0\ // 1 for t mod 2*pi = 0\ || | || | |< 2 | + |< 2/ pi\ | ||sin (t) otherwise | ||sin |t + --| otherwise | \\ / \\ \ 2 / /
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\sin^{2}{\left(t \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\sin^{2}{\left(t + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)$$
// 0 for t mod pi = 0\ // 1 for t mod 2*pi = 0\ || | || | |< 2/ pi\ | + |< 2 | ||cos |t - --| otherwise | ||cos (t) otherwise | \\ \ 2 / / \\ /
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\cos^{2}{\left(t - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\cos^{2}{\left(t \right)} & \text{otherwise} \end{cases}\right)$$
// 0 for t mod pi = 0\ // 1 for t mod 2*pi = 0\ || | || | || 1 | || 1 | |<------------ otherwise | + |<------- otherwise | || 2/ pi\ | || 2 | ||sec |t - --| | ||sec (t) | \\ \ 2 / / \\ /
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{1}{\sec^{2}{\left(t - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{1}{\sec^{2}{\left(t \right)}} & \text{otherwise} \end{cases}\right)$$
2 2
/ 2/t\\ / 2/t pi\\
|-1 + cot |-|| |-1 + tan |- + --||
\ \2// \ \2 4 //
--------------- + --------------------
2 2
/ 2/t\\ / 2/t pi\\
|1 + cot |-|| |1 + tan |- + --||
\ \2// \ \2 4 //
$$\frac{\left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} + \frac{\left(\cot^{2}{\left(\frac{t}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}}$$
// 0 for t mod pi = 0\ // 1 for t mod 2*pi = 0\ || | || | || 1 | || 1 | |<------- otherwise | + |<------------ otherwise | || 2 | || 2/pi \ | ||csc (t) | ||csc |-- - t| | \\ / \\ \2 / /
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{1}{\csc^{2}{\left(t \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{1}{\csc^{2}{\left(- t + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
2 2
/ 2/t pi\\ / 2/t\\
|1 - cot |- + --|| |1 - tan |-||
\ \2 4 // \ \2//
------------------- + --------------
2 2
/ 2/t pi\\ / 2/t\\
|1 + cot |- + --|| |1 + tan |-||
\ \2 4 // \ \2//
$$\frac{\left(- \tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} + \frac{\left(- \cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}{\left(\cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
// / 3*pi\ \
// 1 for t mod 2*pi = 0\ || 1 for |t + ----| mod 2*pi = 0|
|| | || \ 2 / |
|< 2 | + |< |
||cos (t) otherwise | || 4/t\ 2/t\ |
\\ / ||- 4*cos |-| + 4*cos |-| otherwise |
\\ \2/ \2/ /
$$\left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\cos^{2}{\left(t \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \left(t + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\- 4 \cos^{4}{\left(\frac{t}{2} \right)} + 4 \cos^{2}{\left(\frac{t}{2} \right)} & \text{otherwise} \end{cases}\right)$$
// / pi\ \
// 0 for t mod pi = 0\ || 0 for |t + --| mod pi = 0|
|| | || \ 2 / |
|< 2 | + |< |
||sin (t) otherwise | || 2 2/t pi\ |
\\ / ||(1 + sin(t)) *cot |- + --| otherwise |
\\ \2 4 / /
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\sin^{2}{\left(t \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(t + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(t \right)} + 1\right)^{2} \cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)$$
2
/ 4/t\\
| 4*sin |-||
| \2/|
|1 - ---------| 4/t\
| 2 | 16*sin |-|
\ sin (t) / \2/
---------------- + ------------------------
2 2
/ 4/t\\ / 4/t\\
| 4*sin |-|| | 4*sin |-||
| \2/| | \2/| 2
|1 + ---------| |1 + ---------| *sin (t)
| 2 | | 2 |
\ sin (t) / \ sin (t) /
$$\frac{\left(- \frac{4 \sin^{4}{\left(\frac{t}{2} \right)}}{\sin^{2}{\left(t \right)}} + 1\right)^{2}}{\left(\frac{4 \sin^{4}{\left(\frac{t}{2} \right)}}{\sin^{2}{\left(t \right)}} + 1\right)^{2}} + \frac{16 \sin^{4}{\left(\frac{t}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{t}{2} \right)}}{\sin^{2}{\left(t \right)}} + 1\right)^{2} \sin^{2}{\left(t \right)}}$$
// 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 | |<| | + |<| | ||< 2 otherwise | ||< 2 otherwise | |||sin (t) otherwise | |||cos (t) otherwise | \\\ / \\\ /
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\sin^{2}{\left(t \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\cos^{2}{\left(t \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
// 0 for t mod pi = 0\ // 1 for t mod 2*pi = 0\ || | || | || 2/t\ | || 2 | || 4*cot |-| | ||/ 2/t\\ | || \2/ | |||-1 + cot |-|| | |<-------------- otherwise | + |<\ \2// | || 2 | ||--------------- otherwise | ||/ 2/t\\ | || 2 | |||1 + cot |-|| | || / 2/t\\ | ||\ \2// | || |1 + cot |-|| | \\ / \\ \ \2// /
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{t}{2} \right)}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{t}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// 0 for t mod pi = 0\ // 1 for t mod 2*pi = 0\ || | || | || 2/t\ | || 2 | || 4*tan |-| | ||/ 2/t\\ | || \2/ | |||1 - tan |-|| | |<-------------- otherwise | + |<\ \2// | || 2 | ||-------------- otherwise | ||/ 2/t\\ | || 2 | |||1 + tan |-|| | ||/ 2/t\\ | ||\ \2// | |||1 + tan |-|| | \\ / \\\ \2// /
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{4 \tan^{2}{\left(\frac{t}{2} \right)}}{\left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(- \tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// 1 for t mod 2*pi = 0\
|| |
// 0 for t mod pi = 0\ || 2 |
|| | ||/ 1 \ |
|| 4 | |||-1 + -------| |
||---------------------- otherwise | ||| 2/t\| |
|| 2 | ||| tan |-|| |
|
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{4}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \tan^{2}{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{t}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{t}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// / pi\ \
// 0 for t mod pi = 0\ || 0 for |t + --| mod pi = 0|
|| | || \ 2 / |
|| 2/t\ | || |
|| 4*cot |-| | || 2/t pi\ |
|| \2/ | || 4*cot |- + --| |
|<-------------- otherwise | + |< \2 4 / |
|| 2 | ||------------------- otherwise |
||/ 2/t\\ | || 2 |
|||1 + cot |-|| | ||/ 2/t pi\\ |
||\ \2// | |||1 + cot |- + --|| |
\\ / ||\ \2 4 // |
\\ /
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{t}{2} \right)}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(t + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)}}{\left(\cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)$$
2
/ 2/t pi\\
| cos |- - --||
| \2 2 /|
|1 - ------------|
| 2/t\ | 2/t pi\
| cos |-| | 4*cos |- - --|
\ \2/ / \2 2 /
------------------- + ---------------------------
2 2
/ 2/t pi\\ / 2/t pi\\
| cos |- - --|| | cos |- - --||
| \2 2 /| | \2 2 /| 2/t\
|1 + ------------| |1 + ------------| *cos |-|
| 2/t\ | | 2/t\ | \2/
| cos |-| | | cos |-| |
\ \2/ / \ \2/ /
$$\frac{\left(1 - \frac{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} \right)}}\right)^{2}}{\left(1 + \frac{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} \right)}}\right)^{2}} + \frac{4 \cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \cos^{2}{\left(\frac{t}{2} \right)}}$$
2
/ 2/t\ \
| sec |-| |
| \2/ |
|1 - ------------|
| 2/t pi\| 2/t\
| sec |- - --|| 4*sec |-|
\ \2 2 // \2/
------------------- + --------------------------------
2 2
/ 2/t\ \ / 2/t\ \
| sec |-| | | sec |-| |
| \2/ | | \2/ | 2/t pi\
|1 + ------------| |1 + ------------| *sec |- - --|
| 2/t pi\| | 2/t pi\| \2 2 /
| sec |- - --|| | sec |- - --||
\ \2 2 // \ \2 2 //
$$\frac{\left(- \frac{\sec^{2}{\left(\frac{t}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}}{\left(\frac{\sec^{2}{\left(\frac{t}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}} + \frac{4 \sec^{2}{\left(\frac{t}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{t}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}$$
2
/ 2/pi t\\
| csc |-- - -||
| \2 2/|
|1 - ------------|
| 2/t\ | 2/pi t\
| csc |-| | 4*csc |-- - -|
\ \2/ / \2 2/
------------------- + ---------------------------
2 2
/ 2/pi t\\ / 2/pi t\\
| csc |-- - -|| | csc |-- - -||
| \2 2/| | \2 2/| 2/t\
|1 + ------------| |1 + ------------| *csc |-|
| 2/t\ | | 2/t\ | \2/
| csc |-| | | csc |-| |
\ \2/ / \ \2/ /
$$\frac{\left(1 - \frac{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{t}{2} \right)}}\right)^{2}}{\left(1 + \frac{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{t}{2} \right)}}\right)^{2}} + \frac{4 \csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \csc^{2}{\left(\frac{t}{2} \right)}}$$
// / 3*pi\ \
// 1 for t mod 2*pi = 0\ || 1 for |t + ----| mod 2*pi = 0|
|| | || \ 2 / |
|| 2 | || |
||/ 2/t\\ | || 2 |
|||-1 + cot |-|| | ||/ 2/t pi\\ |
|<\ \2// | + |<|-1 + tan |- + --|| |
||--------------- otherwise | ||\ \2 4 // |
|| 2 | ||-------------------- otherwise |
|| / 2/t\\ | || 2 |
|| |1 + cot |-|| | ||/ 2/t pi\\ |
\\ \ \2// / |||1 + tan |- + --|| |
\\\ \2 4 // /
$$\left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{t}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \left(t + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// 0 for t mod pi = 0\
|| | // 1 for t mod 2*pi = 0\
|| 2 | || |
|| sin (t) | || 2 |
||------------------------ otherwise | ||/ 2 4/t\\ |
|| 2 | |||sin (t) - 4*sin |-|| |
|
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{\sin^{2}{\left(t \right)}}{\left(1 + \frac{\sin^{2}{\left(t \right)}}{4 \sin^{4}{\left(\frac{t}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(- 4 \sin^{4}{\left(\frac{t}{2} \right)} + \sin^{2}{\left(t \right)}\right)^{2}}{\left(4 \sin^{4}{\left(\frac{t}{2} \right)} + \sin^{2}{\left(t \right)}\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// 1 for t mod 2*pi = 0\
|| |
// 0 for t mod pi = 0\ || 2 |
|| | ||/ 2 \ |
|| 2 | ||| sin (t) | |
|| sin (t) | |||-1 + ---------| |
||------------------------ otherwise | ||| 4/t\| |
|| 2 | ||| 4*sin |-|| |
|
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{\sin^{2}{\left(t \right)}}{\left(1 + \frac{\sin^{2}{\left(t \right)}}{4 \sin^{4}{\left(\frac{t}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sin^{2}{\left(t \right)}}{4 \sin^{4}{\left(\frac{t}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sin^{2}{\left(t \right)}}{4 \sin^{4}{\left(\frac{t}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// 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 | ||| | ||| | ||| 2/t\ | ||| 2 | ||| 4*cot |-| | |||/ 2/t\\ | |<| \2/ | + |<||-1 + cot |-|| | ||<-------------- otherwise otherwise | ||<\ \2// otherwise | ||| 2 | |||--------------- otherwise | |||/ 2/t\\ | ||| 2 | ||||1 + cot |-|| | ||| / 2/t\\ | |||\ \2// | ||| |1 + cot |-|| | \\\ / \\\ \ \2// /
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{t}{2} \right)}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{t}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
// 1 for t mod 2*pi = 0\
|| |
// 0 for t mod pi = 0\ || 2 |
|| | ||/ 2/t\ \ |
|| 2/t\ | ||| cos |-| | |
|| 4*cos |-| | ||| \2/ | |
|| \2/ | |||-1 + ------------| |
||-------------------------------- otherwise | ||| 2/t pi\| |
|| 2 | ||| cos |- - --|| |
|
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{4 \cos^{2}{\left(\frac{t}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{t}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(\frac{\cos^{2}{\left(\frac{t}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\cos^{2}{\left(\frac{t}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// 1 for t mod 2*pi = 0\
|| |
// 0 for t mod pi = 0\ || 2 |
|| | ||/ 2/t pi\\ |
|| 2/t pi\ | ||| sec |- - --|| |
|| 4*sec |- - --| | ||| \2 2 /| |
|| \2 2 / | |||-1 + ------------| |
||--------------------------- otherwise | ||| 2/t\ | |
|| 2 | ||| sec |-| | |
|
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{4 \sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \sec^{2}{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// 1 for t mod 2*pi = 0\
|| |
// 0 for t mod pi = 0\ || 2 |
|| | ||/ 2/t\ \ |
|| 2/t\ | ||| csc |-| | |
|| 4*csc |-| | ||| \2/ | |
|| \2/ | |||-1 + ------------| |
||-------------------------------- otherwise | ||| 2/pi t\| |
|| 2 | ||| csc |-- - -|| |
|
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{4 \csc^{2}{\left(\frac{t}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{t}{2} \right)}}{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2} \csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{t}{2} \right)}}{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\csc^{2}{\left(\frac{t}{2} \right)}}{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)$$
Piecewise((0, Mod(t = pi, 0)), (4*csc(t/2)^2/((1 + csc(t/2)^2/csc(pi/2 - t/2)^2)^2*csc(pi/2 - t/2)^2), True)) + Piecewise((1, Mod(t = 2*pi, 0)), ((-1 + csc(t/2)^2/csc(pi/2 - t/2)^2)^2/(1 + csc(t/2)^2/csc(pi/2 - t/2)^2)^2, True))