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