Господин Экзамен
Другие калькуляторы
-
Как пользоваться?
- b^2+11*b+25 если b=-4
- a^52*a^23 если a=-3/2
- tan(t)^2*(-1+1/(sin(t)^2)) если t=1/2
- x если x=3
- Разложить многочлен на множители:
- x^2-y^2+x-y
- 9*x^2-24*x*y+16*y^2
- 4*x^2+25
- X^2-4
- Общий знаменатель:
- 7/(d*x-d*y)+10/(n*y-n*x)
- n/(k-5)^2+10/(5-k)^2
- n/(p-9)+18/(9-p)
- 2/(2-sqrt(7))+3/(2+sqrt(7))+sqrt(7)/3
- Умножение столбиком:
- 76*57
- 30*35
- 19*35
- 80*15
- Деление столбиком:
- 2738/9
- 141/2
- 5108/7
- 38448/2
- Раскрыть скобки в:
- 18*c^2-2*(3*c-1)^2
- 2*(x+y)
- 7*(c-2)-10
- 25*n^2-(7+5*n)*(7-5*n)
- Разложение числа на простые множители:
- 4608
- 5436
- 515
- 74
- График функции y =:
- 1/2
- Производная:
- 1/2
- Интеграл d{x}:
-
1/2
-
Идентичные выражения
- tan(t)^ два *(- один + один /(sin(t)^ два)) если t= один / два
- тангенс от (t) в квадрате умножить на ( минус 1 плюс 1 делить на ( синус от (t) в квадрате )) если t равно 1 делить на 2
- тангенс от (t) в степени два умножить на ( минус один плюс один делить на ( синус от (t) в степени два)) если t равно один делить на два
- tan(t)2*(-1+1/(sin(t)2)) если t=1/2
- tant2*-1+1/sint2 если t=1/2
- tan(t)²*(-1+1/(sin(t)²)) если t=1/2
- tan(t) в степени 2*(-1+1/(sin(t) в степени 2)) если t=1/2
- tan(t)^2(-1+1/(sin(t)^2)) если t=1/2
- tan(t)2(-1+1/(sin(t)2)) если t=1/2
- tant2-1+1/sint2 если t=1/2
- tant^2-1+1/sint^2 если t=1/2
- tan(t)^2*(-1+1/(sin(t)^2)) при t=1/2
- tan(t)^2*(-1+1 разделить на (sin(t)^2)) если t=1 разделить на 2
-
Похожие выражения
- tan(t)^2*(1+1/(sin(t)^2)) если t=1/2
- tan(t)^2*(-1-1/(sin(t)^2)) если t=1/2
tan(t)^2*(-1+1/(sin(t)^2)) если t=1/2
Решение
Вы ввели
[src]
2 / 1 \
tan (t)*|-1 + 1*-------|
| 2 |
\ sin (t)/
$$\left(-1 + 1 \cdot \frac{1}{\sin^{2}{\left(t \right)}}\right) \tan^{2}{\left(t \right)}$$
tan(t)^2*(-1 + 1/sin(t)^2)
Комбинаторика
[src]
2
-tan (t)*(1 + sin(t))*(-1 + sin(t))
------------------------------------
2
sin (t)
$$- \frac{\left(\sin{\left(t \right)} - 1\right) \left(\sin{\left(t \right)} + 1\right) \tan^{2}{\left(t \right)}}{\sin^{2}{\left(t \right)}}$$
-tan(t)^2*(1 + sin(t))*(-1 + sin(t))/sin(t)^2
Общий знаменатель
[src]
2
2 tan (t)
- tan (t) + -------
2
sin (t)
$$- \tan^{2}{\left(t \right)} + \frac{\tan^{2}{\left(t \right)}}{\sin^{2}{\left(t \right)}}$$
-tan(t)^2 + tan(t)^2/sin(t)^2
Рациональный знаменатель
[src]
2
2 tan (t)
- tan (t) + -------
2
sin (t)
$$- \tan^{2}{\left(t \right)} + \frac{\tan^{2}{\left(t \right)}}{\sin^{2}{\left(t \right)}}$$
2 / 2 \
tan (t)*\1 - sin (t)/
---------------------
2
sin (t)
$$\frac{\left(- \sin^{2}{\left(t \right)} + 1\right) \tan^{2}{\left(t \right)}}{\sin^{2}{\left(t \right)}}$$
tan(t)^2*(1 - sin(t)^2)/sin(t)^2
Степени
[src]
2 / 1 \
tan (t)*|-1 + -------|
| 2 |
\ sin (t)/
$$\left(-1 + \frac{1}{\sin^{2}{\left(t \right)}}\right) \tan^{2}{\left(t \right)}$$
2
/ I*t -I*t\ / 4 \
-\- e + e / *|-1 - -----------------|
| 2|
| / -I*t I*t\ |
\ \- e + e / /
--------------------------------------------
2
/ I*t -I*t\
\e + e /
$$- \frac{\left(-1 - \frac{4}{\left(e^{i t} - e^{- i t}\right)^{2}}\right) \left(- e^{i t} + e^{- i t}\right)^{2}}{\left(e^{i t} + e^{- i t}\right)^{2}}$$
-(-exp(i*t) + exp(-i*t))^2*(-1 - 4/(-exp(-i*t) + exp(i*t))^2)/(exp(i*t) + exp(-i*t))^2
Объединение рациональных выражений
[src]
2 / 2 \
tan (t)*\1 - sin (t)/
---------------------
2
sin (t)
$$\frac{\left(- \sin^{2}{\left(t \right)} + 1\right) \tan^{2}{\left(t \right)}}{\sin^{2}{\left(t \right)}}$$
tan(t)^2*(1 - sin(t)^2)/sin(t)^2
Собрать выражение
[src]
2 / 2 \ tan (t)*\-1 + csc (t)/
$$\left(\csc^{2}{\left(t \right)} - 1\right) \tan^{2}{\left(t \right)}$$
2 / 1 \
tan (t)*|-1 + -------|
| 2 |
\ sin (t)/
$$\left(-1 + \frac{1}{\sin^{2}{\left(t \right)}}\right) \tan^{2}{\left(t \right)}$$
tan(t)^2*(-1 + sin(t)^(-2))
Тригонометрическая часть
[src]
1
$$1$$
2 / 1 \
tan (t)*|-1 + -------|
| 2 |
\ sin (t)/
$$\left(-1 + \frac{1}{\sin^{2}{\left(t \right)}}\right) \tan^{2}{\left(t \right)}$$
2 / 2 \
sec (t)*\-1 + csc (t)/
----------------------
2
csc (t)
$$\frac{\left(\csc^{2}{\left(t \right)} - 1\right) \sec^{2}{\left(t \right)}}{\csc^{2}{\left(t \right)}}$$
2 / 2 \
tan (t)*\1 - sin (t)/
---------------------
2
sin (t)
$$\frac{\left(- \sin^{2}{\left(t \right)} + 1\right) \tan^{2}{\left(t \right)}}{\sin^{2}{\left(t \right)}}$$
2 / 2 \
4*csc (2*t)*\-1 + csc (t)/
--------------------------
4
csc (t)
$$\frac{4 \left(\csc^{2}{\left(t \right)} - 1\right) \csc^{2}{\left(2 t \right)}}{\csc^{4}{\left(t \right)}}$$
2 / 1 \
sin (t)*|-1 + -------|
| 2 |
\ sin (t)/
----------------------
2
cos (t)
$$\frac{\left(-1 + \frac{1}{\sin^{2}{\left(t \right)}}\right) \sin^{2}{\left(t \right)}}{\cos^{2}{\left(t \right)}}$$
4 / 1 \
4*sin (t)*|-1 + -------|
| 2 |
\ sin (t)/
------------------------
2
sin (2*t)
$$\frac{4 \left(-1 + \frac{1}{\sin^{2}{\left(t \right)}}\right) \sin^{4}{\left(t \right)}}{\sin^{2}{\left(2 t \right)}}$$
2/pi \ / 2 \
csc |-- - t|*\-1 + csc (t)/
\2 /
---------------------------
2
csc (t)
$$\frac{\left(\csc^{2}{\left(t \right)} - 1\right) \csc^{2}{\left(- t + \frac{\pi}{2} \right)}}{\csc^{2}{\left(t \right)}}$$
2 / 1 \
sin (t)*|-1 + -------|
| 2 |
\ sin (t)/
----------------------
2/ pi\
sin |t + --|
\ 2 /
$$\frac{\left(-1 + \frac{1}{\sin^{2}{\left(t \right)}}\right) \sin^{2}{\left(t \right)}}{\sin^{2}{\left(t + \frac{\pi}{2} \right)}}$$
2/t\
4*tan |-|
\2/
----------------------
2
/ 2/t\\ 2
|1 - tan |-|| *tan (t)
\ \2//
$$\frac{4 \tan^{2}{\left(\frac{t}{2} \right)}}{\left(- \tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2} \tan^{2}{\left(t \right)}}$$
2 / 2/ pi\\
sec (t)*|-1 + sec |t - --||
\ \ 2 //
---------------------------
2/ pi\
sec |t - --|
\ 2 /
$$\frac{\left(\sec^{2}{\left(t - \frac{\pi}{2} \right)} - 1\right) \sec^{2}{\left(t \right)}}{\sec^{2}{\left(t - \frac{\pi}{2} \right)}}$$
2/pi \ / 2 \
csc |-- - t|*\-1 + csc (pi - t)/
\2 /
--------------------------------
2
csc (pi - t)
$$\frac{\left(\csc^{2}{\left(- t + \pi \right)} - 1\right) \csc^{2}{\left(- t + \frac{\pi}{2} \right)}}{\csc^{2}{\left(- t + \pi \right)}}$$
2 / 2/pi \\
sec (t)*|-1 + sec |-- - t||
\ \2 //
---------------------------
2/pi \
sec |-- - t|
\2 /
$$\frac{\left(\sec^{2}{\left(- t + \frac{\pi}{2} \right)} - 1\right) \sec^{2}{\left(t \right)}}{\sec^{2}{\left(- t + \frac{\pi}{2} \right)}}$$
/ 2\
| / 2/t\\ |
| |1 + tan |-|| |
2 | \ \2// |
tan (t)*|-1 + --------------|
| 2/t\ |
| 4*tan |-| |
\ \2/ /
$$\left(\frac{\left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}}{4 \tan^{2}{\left(\frac{t}{2} \right)}} - 1\right) \tan^{2}{\left(t \right)}$$
2/ pi\ / 1 \
cos |t - --|*|-1 + ------------|
\ 2 / | 2/ pi\|
| cos |t - --||
\ \ 2 //
--------------------------------
2
cos (t)
$$\frac{\left(-1 + \frac{1}{\cos^{2}{\left(t - \frac{\pi}{2} \right)}}\right) \cos^{2}{\left(t - \frac{\pi}{2} \right)}}{\cos^{2}{\left(t \right)}}$$
2
/ 1 \ 2/t\
|1 + -------| *cot |-|
| 2/t\| \2/
| cot |-||
\ \2//
-1 + ----------------------
4
---------------------------
2
cot (t)
$$\frac{\frac{\left(1 + \frac{1}{\cot^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \cot^{2}{\left(\frac{t}{2} \right)}}{4} - 1}{\cot^{2}{\left(t \right)}}$$
2/ pi\ / 2/ pi\\
4*sec |2*t - --|*|-1 + sec |t - --||
\ 2 / \ \ 2 //
------------------------------------
4/ pi\
sec |t - --|
\ 2 /
$$\frac{4 \left(\sec^{2}{\left(t - \frac{\pi}{2} \right)} - 1\right) \sec^{2}{\left(2 t - \frac{\pi}{2} \right)}}{\sec^{4}{\left(t - \frac{\pi}{2} \right)}}$$
4/ pi\ / 1 \
4*cos |t - --|*|-1 + ------------|
\ 2 / | 2/ pi\|
| cos |t - --||
\ \ 2 //
----------------------------------
2/ pi\
cos |2*t - --|
\ 2 /
$$\frac{4 \left(-1 + \frac{1}{\cos^{2}{\left(t - \frac{\pi}{2} \right)}}\right) \cos^{4}{\left(t - \frac{\pi}{2} \right)}}{\cos^{2}{\left(2 t - \frac{\pi}{2} \right)}}$$
2 / 1 \
sin (t)*|-1 + ---------------------|
| 2 2/t\|
| (1 + cos(t)) *tan |-||
\ \2//
------------------------------------
2
cos (t)
$$\frac{\left(-1 + \frac{1}{\left(\cos{\left(t \right)} + 1\right)^{2} \tan^{2}{\left(\frac{t}{2} \right)}}\right) \sin^{2}{\left(t \right)}}{\cos^{2}{\left(t \right)}}$$
/ // zoo for t mod pi = 0\\
| || ||
2 | || 1 ||
tan (t)*|-1 + |<------- otherwise ||
| || 2 ||
| ||sin (t) ||
\ \\ //
$$\left(\left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\frac{1}{\sin^{2}{\left(t \right)}} & \text{otherwise} \end{cases}\right) - 1\right) \tan^{2}{\left(t \right)}$$
/ / zoo for t mod pi = 0\
| | |
| | 1 |
| <------- otherwise |
| | 2 |
| |sin (t) |
2 | 1 \ |
4*tan (t)*|- - + --------------------------|
\ 4 4 /
$$4 \left(\left(\frac{\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\frac{1}{\sin^{2}{\left(t \right)}} & \text{otherwise} \end{cases}}{4}\right) - \frac{1}{4}\right) \tan^{2}{\left(t \right)}$$
/ // zoo for t mod pi = 0\\
| || ||
4 | || 1 ||
4*sin (t)*|-1 + |<------- otherwise ||
| || 2 ||
| ||sin (t) ||
\ \\ //
---------------------------------------------
2
sin (2*t)
$$\frac{4 \left(\left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\frac{1}{\sin^{2}{\left(t \right)}} & \text{otherwise} \end{cases}\right) - 1\right) \sin^{4}{\left(t \right)}}{\sin^{2}{\left(2 t \right)}}$$
/ 2\
| / 2/t\\ |
| |1 + tan |-|| |
2/t\ | \ \2// |
4*tan |-|*|-1 + --------------|
\2/ | 2/t\ |
| 4*tan |-| |
\ \2/ /
-------------------------------
2
/ 2/t\\
|1 - tan |-||
\ \2//
$$\frac{4 \left(\frac{\left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}}{4 \tan^{2}{\left(\frac{t}{2} \right)}} - 1\right) \tan^{2}{\left(\frac{t}{2} \right)}}{\left(- \tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}}$$
/ // zoo for t mod pi = 0\\
2/pi \ | || ||
csc |-- - t|*|-1 + |< 2 ||
\2 / | ||csc (t) otherwise ||
\ \\ //
------------------------------------------------
2
csc (t)
$$\frac{\left(\left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\csc^{2}{\left(t \right)} & \text{otherwise} \end{cases}\right) - 1\right) \csc^{2}{\left(- t + \frac{\pi}{2} \right)}}{\csc^{2}{\left(t \right)}}$$
2 2
(1 - cos(t) + sin(t)) *(-1 + cos(t) + sin(t))
----------------------------------------------
2
2*(1 - cos(t)) *(1 + cos(2*t))
$$\frac{\left(\sin{\left(t \right)} - \cos{\left(t \right)} + 1\right)^{2} \left(\sin{\left(t \right)} + \cos{\left(t \right)} - 1\right)^{2}}{2 \left(- \cos{\left(t \right)} + 1\right)^{2} \left(\cos{\left(2 t \right)} + 1\right)}$$
/ // zoo for t mod pi = 0\\
| || ||
| || 2/t\ ||
2 | || tan |-| ||
tan (t)*|-1 + |< \2/ ||
| ||-------------- otherwise ||
| || 2 ||
| ||(-1 + cos(t)) ||
\ \\ //
$$\left(\left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\frac{\tan^{2}{\left(\frac{t}{2} \right)}}{\left(\cos{\left(t \right)} - 1\right)^{2}} & \text{otherwise} \end{cases}\right) - 1\right) \tan^{2}{\left(t \right)}$$
/ // zoo for t mod pi = 0\\
2 | || ||
sec (t)*|-1 + |< 2/ pi\ ||
| ||sec |t - --| otherwise ||
\ \\ \ 2 / //
------------------------------------------------
2/ pi\
sec |t - --|
\ 2 /
$$\frac{\left(\left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\sec^{2}{\left(t - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) - 1\right) \sec^{2}{\left(t \right)}}{\sec^{2}{\left(t - \frac{\pi}{2} \right)}}$$
2
/ 2/t\\
|1 + tan |-|| *(1 - cos(2*t))
\ \2//
--------------------------------------
2 2
/ /t\\ / /t\\ 2
2*|1 + tan|-|| *|-1 + tan|-|| *tan (t)
\ \2// \ \2//
$$\frac{\left(- \cos{\left(2 t \right)} + 1\right) \left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}}{2 \left(\tan{\left(\frac{t}{2} \right)} - 1\right)^{2} \left(\tan{\left(\frac{t}{2} \right)} + 1\right)^{2} \tan^{2}{\left(t \right)}}$$
// zoo for t mod pi = 0\
|| |
|| 2 |
||/ 2/t\\ |
|||1 + cot |-|| |
-1 + |<\ \2// |
||-------------- otherwise |
|| 2/t\ |
|| 4*cot |-| |
|| \2/ |
\\ /
----------------------------------------
2
cot (t)
$$\frac{\left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}}{4 \cot^{2}{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}\right) - 1}{\cot^{2}{\left(t \right)}}$$
/ 2 \
| / 4/t\\ |
| | 4*sin |-|| |
| | \2/| 2 |
| |1 + ---------| *sin (t)|
| | 2 | |
4 | \ sin (t) / |
4*sin (t)*|-1 + ------------------------|
| 4/t\ |
| 16*sin |-| |
\ \2/ /
-----------------------------------------
2
sin (2*t)
$$\frac{4 \left(\frac{\left(\frac{4 \sin^{4}{\left(\frac{t}{2} \right)}}{\sin^{2}{\left(t \right)}} + 1\right)^{2} \sin^{2}{\left(t \right)}}{16 \sin^{4}{\left(\frac{t}{2} \right)}} - 1\right) \sin^{4}{\left(t \right)}}{\sin^{2}{\left(2 t \right)}}$$
/ // zoo for t mod pi = 0\\
| || ||
2/ pi\ | || 1 ||
cos |t - --|*|-1 + |<------------ otherwise ||
\ 2 / | || 2/ pi\ ||
| ||cos |t - --| ||
\ \\ \ 2 / //
-----------------------------------------------------
2
cos (t)
$$\frac{\left(\left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\frac{1}{\cos^{2}{\left(t - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) - 1\right) \cos^{2}{\left(t - \frac{\pi}{2} \right)}}{\cos^{2}{\left(t \right)}}$$
/ // zoo for t mod pi = 0\\
| || ||
| || 2 ||
| ||/ 2/t\\ ||
2 | |||1 + tan |-|| ||
tan (t)*|-1 + |<\ \2// ||
| ||-------------- otherwise ||
| || 2/t\ ||
| || 4*tan |-| ||
| || \2/ ||
\ \\ //
$$\left(\left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\frac{\left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}}{4 \tan^{2}{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}\right) - 1\right) \tan^{2}{\left(t \right)}$$
/ // zoo for t mod pi = 0\\
| || ||
| || 2 ||
| ||/ 1 \ 2/t\ ||
2 | |||1 + -------| *tan |-| ||
tan (t)*|-1 + |<| 2/t\| \2/ ||
| ||| tan |-|| ||
| ||\ \2// ||
| ||---------------------- otherwise ||
| || 4 ||
\ \\ //
$$\left(\left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\frac{\left(1 + \frac{1}{\tan^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \tan^{2}{\left(\frac{t}{2} \right)}}{4} & \text{otherwise} \end{cases}\right) - 1\right) \tan^{2}{\left(t \right)}$$
/ // zoo for t mod pi = 0\\
| || ||
| ||/ zoo for t mod pi = 0 ||
2 | ||| ||
tan (t)*|-1 + |<| 1 ||
| ||<------- otherwise otherwise ||
| ||| 2 ||
| |||sin (t) ||
\ \\\ //
$$\left(\left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\frac{1}{\sin^{2}{\left(t \right)}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) - 1\right) \tan^{2}{\left(t \right)}$$
/ // zoo for t mod pi = 0\\
| || ||
| || 2/t\ / 1 \ ||
2 | ||-2*sin |-|*|1 + -------| ||
tan (t)*|-1 + |< \2/ | 2 | ||
| || \ tan (t)/ ||
| ||------------------------ otherwise ||
| || -1 + cos(t) ||
\ \\ //
$$\left(\left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\- \frac{2 \cdot \left(1 + \frac{1}{\tan^{2}{\left(t \right)}}\right) \sin^{2}{\left(\frac{t}{2} \right)}}{\cos{\left(t \right)} - 1} & \text{otherwise} \end{cases}\right) - 1\right) \tan^{2}{\left(t \right)}$$
/ 2\
| / 2/t\\ |
2 | |1 + tan |-|| |
/ 2 \ 4/t\ | \ \2// |
16*\1 + tan (t)/ *tan |-|*|-1 + --------------|
\2/ | 2/t\ |
| 4*tan |-| |
\ \2/ /
-----------------------------------------------
4
/ 2/t\\ 2
|1 + tan |-|| *tan (t)
\ \2//
$$\frac{16 \left(\frac{\left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}}{4 \tan^{2}{\left(\frac{t}{2} \right)}} - 1\right) \left(\tan^{2}{\left(t \right)} + 1\right)^{2} \tan^{4}{\left(\frac{t}{2} \right)}}{\left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{4} \tan^{2}{\left(t \right)}}$$
/ 2 \
| / 2/t pi\\ |
| | cos |- - --|| |
| | \2 2 /| 2/t\|
| |1 + ------------| *cos |-||
| | 2/t\ | \2/|
| | cos |-| | |
2/ pi\ | \ \2/ / |
cos |t - --|*|-1 + ---------------------------|
\ 2 / | 2/t pi\ |
| 4*cos |- - --| |
\ \2 2 / /
-----------------------------------------------
2
cos (t)
$$\frac{\left(\frac{\left(1 + \frac{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \cos^{2}{\left(\frac{t}{2} \right)}}{4 \cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} - 1\right) \cos^{2}{\left(t - \frac{\pi}{2} \right)}}{\cos^{2}{\left(t \right)}}$$
/ 2 \
| / 2/t\ \ |
| | sec |-| | |
| | \2/ | 2/t pi\|
| |1 + ------------| *sec |- - --||
| | 2/t pi\| \2 2 /|
| | sec |- - --|| |
2 | \ \2 2 // |
sec (t)*|-1 + --------------------------------|
| 2/t\ |
| 4*sec |-| |
\ \2/ /
-----------------------------------------------
2/ pi\
sec |t - --|
\ 2 /
$$\frac{\left(\frac{\left(\frac{\sec^{2}{\left(\frac{t}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{4 \sec^{2}{\left(\frac{t}{2} \right)}} - 1\right) \sec^{2}{\left(t \right)}}{\sec^{2}{\left(t - \frac{\pi}{2} \right)}}$$
/ 2 \
| / 2/pi t\\ |
| | csc |-- - -|| |
| | \2 2/| 2/t\|
| |1 + ------------| *csc |-||
| | 2/t\ | \2/|
| | csc |-| | |
2/pi \ | \ \2/ / |
csc |-- - t|*|-1 + ---------------------------|
\2 / | 2/pi t\ |
| 4*csc |-- - -| |
\ \2 2/ /
-----------------------------------------------
2
csc (t)
$$\frac{\left(\frac{\left(1 + \frac{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \csc^{2}{\left(\frac{t}{2} \right)}}{4 \csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}} - 1\right) \csc^{2}{\left(- t + \frac{\pi}{2} \right)}}{\csc^{2}{\left(t \right)}}$$
// zoo for t mod pi = 0\
|| |
||/ zoo for t mod pi = 0 |
||| |
||| 2 |
|||/ 2/t\\ |
-1 + |<||1 + cot |-|| |
||<\ \2// otherwise |
|||-------------- otherwise |
||| 2/t\ |
||| 4*cot |-| |
||| \2/ |
\\\ /
-----------------------------------------------------------
2
cot (t)
$$\frac{\left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}}{4 \cot^{2}{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) - 1}{\cot^{2}{\left(t \right)}}$$
/ 2\
| / 2/t\\ |
2 | |1 + cot |-|| |
/ 2/t pi\\ 2/t\ | \ \2// |
|1 + tan |- + --|| *cot |-|*|-1 + --------------|
\ \2 4 // \2/ | 2/t\ |
| 4*cot |-| |
\ \2/ /
-------------------------------------------------
2
/ 2/t\\ 2/t pi\
|1 + cot |-|| *tan |- + --|
\ \2// \2 4 /
$$\frac{\left(\frac{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}}{4 \cot^{2}{\left(\frac{t}{2} \right)}} - 1\right) \left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \cot^{2}{\left(\frac{t}{2} \right)}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2} \tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)}}$$
/ // zoo for t mod pi = 0\\
| || ||
| || 2 ||
| ||/ 2 \ ||
| ||| sin (t) | 4/t\ ||
4 | |||1 + ---------| *sin |-| ||
4*sin (t)*|-1 + |<| 4/t\| \2/ ||
| ||| 4*sin |-|| ||
| ||\ \2// ||
| ||------------------------ otherwise ||
| || 2 ||
| || sin (t) ||
\ \\ //
--------------------------------------------------------------
2
sin (2*t)
$$\frac{4 \left(\left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\frac{\left(1 + \frac{\sin^{2}{\left(t \right)}}{4 \sin^{4}{\left(\frac{t}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{t}{2} \right)}}{\sin^{2}{\left(t \right)}} & \text{otherwise} \end{cases}\right) - 1\right) \sin^{4}{\left(t \right)}}{\sin^{2}{\left(2 t \right)}}$$
/ / zoo for t mod pi = 0\
| | |
| | 4/t\ |
| | sin |-| 2 |
| <1 \2/ sin (t) |
| |- + ------- + ---------- otherwise |
| |2 2 4/t\ |
| | sin (t) 16*sin |-| |
/ 1 \ | 1 \ \2/ |
4*|-1 + -------|*|- - + -------------------------------------------|
| 2 | \ 4 4 /
\ cos (t)/
$$4 \left(-1 + \frac{1}{\cos^{2}{\left(t \right)}}\right) \left(\left(\frac{\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\frac{\sin^{4}{\left(\frac{t}{2} \right)}}{\sin^{2}{\left(t \right)}} + \frac{1}{2} + \frac{\sin^{2}{\left(t \right)}}{16 \sin^{4}{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}}{4}\right) - \frac{1}{4}\right)$$
2
/ 2/t pi\\ 2 / 4 \
|1 - cot |- + --|| *(1 + sin(t)) *|-1 + ---------------------------------|
\ \2 4 // | 2 |
| / 2/t pi\\ 2|
| |1 - cot |- + --|| *(1 + sin(t)) |
\ \ \2 4 // /
--------------------------------------------------------------------------
2
/ 2/t\\ 4/t\
4*|1 - tan |-|| *cos |-|
\ \2// \2/
$$\frac{\left(-1 + \frac{4}{\left(- \cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\sin{\left(t \right)} + 1\right)^{2}}\right) \left(- \cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\sin{\left(t \right)} + 1\right)^{2}}{4 \left(- \tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2} \cos^{4}{\left(\frac{t}{2} \right)}}$$
/ // zoo for t mod pi = 0\\
| || ||
| || 2 ||
| ||/ 2/t\ \ ||
| ||| cos |-| | ||
| ||| \2/ | 2/t pi\ ||
2/ pi\ | |||1 + ------------| *cos |- - --| ||
cos |t - --|*|-1 + |<| 2/t pi\| \2 2 / ||
\ 2 / | ||| cos |- - --|| ||
| ||\ \2 2 // ||
| ||-------------------------------- otherwise ||
| || 2/t\ ||
| || 4*cos |-| ||
| || \2/ ||
\ \\ //
-------------------------------------------------------------------------
2
cos (t)
$$\frac{\left(\left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\frac{\left(\frac{\cos^{2}{\left(\frac{t}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{4 \cos^{2}{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}\right) - 1\right) \cos^{2}{\left(t - \frac{\pi}{2} \right)}}{\cos^{2}{\left(t \right)}}$$
/ // zoo for t mod pi = 0\\
| || ||
| || 2 ||
| ||/ 2/t pi\\ ||
| ||| sec |- - --|| ||
| ||| \2 2 /| 2/t\ ||
2 | |||1 + ------------| *sec |-| ||
sec (t)*|-1 + |<| 2/t\ | \2/ ||
| ||| sec |-| | ||
| ||\ \2/ / ||
| ||--------------------------- otherwise ||
| || 2/t pi\ ||
| || 4*sec |- - --| ||
| || \2 2 / ||
\ \\ //
---------------------------------------------------------------
2/ pi\
sec |t - --|
\ 2 /
$$\frac{\left(\left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\frac{\left(1 + \frac{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \sec^{2}{\left(\frac{t}{2} \right)}}{4 \sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) - 1\right) \sec^{2}{\left(t \right)}}{\sec^{2}{\left(t - \frac{\pi}{2} \right)}}$$
/ // zoo for t mod pi = 0\\
| || ||
| || 2 ||
| ||/ 2/t\ \ ||
| ||| csc |-| | ||
| ||| \2/ | 2/pi t\ ||
2/pi \ | |||1 + ------------| *csc |-- - -| ||
csc |-- - t|*|-1 + |<| 2/pi t\| \2 2/ ||
\2 / | ||| csc |-- - -|| ||
| ||\ \2 2// ||
| ||-------------------------------- otherwise ||
| || 2/t\ ||
| || 4*csc |-| ||
| || \2/ ||
\ \\ //
-------------------------------------------------------------------------
2
csc (t)
$$\frac{\left(\left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{t}{2} \right)}}{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2} \csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}}{4 \csc^{2}{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}\right) - 1\right) \csc^{2}{\left(- t + \frac{\pi}{2} \right)}}{\csc^{2}{\left(t \right)}}$$
/ 2 \
| / 2/t pi\\ |
2 2 | |1 + tan |- + --|| |
/ 2/t\\ / 2/t pi\\ | \ \2 4 // |
|1 + cot |-|| *|-1 + tan |- + --|| *|-1 + --------------------|
\ \2// \ \2 4 // | 2|
| / 2/t pi\\ |
| |-1 + tan |- + --|| |
\ \ \2 4 // /
---------------------------------------------------------------
2 2
/ 2/t pi\\ / 2/t\\
|1 + tan |- + --|| *|-1 + cot |-||
\ \2 4 // \ \2//
$$\frac{\left(-1 + \frac{\left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}\right) \left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} - 1\right)^{2} \left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\cot^{2}{\left(\frac{t}{2} \right)} - 1\right)^{2}}$$
/ 2\
| / 2/t pi\\ |
2 2 | |1 + cot |- + --|| |
/ 2/t\\ / 2/t pi\\ | \ \2 4 // |
|1 + tan |-|| *|1 - cot |- + --|| *|-1 + -------------------|
\ \2// \ \2 4 // | 2|
| / 2/t pi\\ |
| |1 - cot |- + --|| |
\ \ \2 4 // /
-------------------------------------------------------------
2 2
/ 2/t pi\\ / 2/t\\
|1 + cot |- + --|| *|1 - tan |-||
\ \2 4 // \ \2//
$$\frac{\left(-1 + \frac{\left(\cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}{\left(- \cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}\right) \left(- \cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}}{\left(- \tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2} \left(\cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
/ // zoo for t mod pi = 0\\ | || || // 0 for t mod pi = 0\ // zoo for 2*t mod pi = 0\ | || 1 || || | || | 4*|-1 + |<------- otherwise ||*|< 4/t\ 8/t\ |*|< 2 | | || 2 || ||16*cot |-|*sin |-| otherwise | ||------------ otherwise | | ||sin (t) || \\ \2/ \2/ / \\1 - cos(4*t) / \ \\ //
$$4 \left(\left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\frac{1}{\sin^{2}{\left(t \right)}} & \text{otherwise} \end{cases}\right) - 1\right) \left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\16 \sin^{8}{\left(\frac{t}{2} \right)} \cot^{4}{\left(\frac{t}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: 2 t \bmod \pi = 0 \\\frac{2}{- \cos{\left(4 t \right)} + 1} & \text{otherwise} \end{cases}\right)$$
/ // / 3*pi\ \\ | || 1 for |t + ----| mod 2*pi = 0|| | || \ 2 / || | || || // / 3*pi\ \ | || 1 || // 1 for t mod 2*pi = 0\ || 1 for |t + ----| mod 2*pi = 0| | ||-1 + ------- || || | || \ 2 / | |-1 + |< 2/t\ ||*|< 2 |*|< | | || sin |-| || ||------------ otherwise | || 4/t\ 2/t\ | | || \2/ || \\1 + cos(2*t) / ||- 4*cos |-| + 4*cos |-| otherwise | | ||------------ otherwise || \\ \2/ \2/ / | || 4/t\ || | || 4*cos |-| || \ \\ \2/ //
$$\left(\left(\begin{cases} 1 & \text{for}\: \left(t + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\sin^{2}{\left(\frac{t}{2} \right)}}}{4 \cos^{4}{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}\right) - 1\right) \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{2}{\cos{\left(2 t \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(t + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\- 4 \cos^{4}{\left(\frac{t}{2} \right)} + 4 \cos^{2}{\left(\frac{t}{2} \right)} & \text{otherwise} \end{cases}\right)$$
/ // zoo for t mod pi = 0\\ // 0 for t mod pi = 0\ | || || || | // zoo for 2*t mod pi = 0\ | || 2 || || 4/t\ | || | | ||/ 2/t\\ || || 16*cot |-| | || 2 | | |||1 + cot |-|| || || \2/ | ||/ 2 \ | 4*|-1 + |<\ \2// ||*|<-------------- otherwise |*|<\1 + cot (t)/ | | ||-------------- otherwise || || 4 | ||-------------- otherwise | | || 2/t\ || ||/ 2/t\\ | || 2 | | || 4*cot |-| || |||1 + cot |-|| | || 4*cot (t) | | || \2/ || ||\ \2// | \\ / \ \\ // \\ /
$$4 \left(\left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}}{4 \cot^{2}{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}\right) - 1\right) \left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{16 \cot^{4}{\left(\frac{t}{2} \right)}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{4}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: 2 t \bmod \pi = 0 \\\frac{\left(\cot^{2}{\left(t \right)} + 1\right)^{2}}{4 \cot^{2}{\left(t \right)}} & \text{otherwise} \end{cases}\right)$$
/ // / 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| | || \ 2 / || || | || \ 2 / | | || || || 2 | || | | || 2 || || / 2/t\\ | || 2 | | ||/ 2/t pi\\ || || |1 + cot |-|| | ||/ 2/t pi\\ | |-1 + |<|1 + tan |- + --|| ||*|< \ \2// |*|<|-1 + tan |- + --|| | | ||\ \2 4 // || ||--------------- otherwise | ||\ \2 4 // | | ||-------------------- otherwise || || 2 | ||-------------------- otherwise | | || 2 || ||/ 2/t\\ | || 2 | | ||/ 2/t pi\\ || |||-1 + cot |-|| | ||/ 2/t pi\\ | | |||-1 + tan |- + --|| || \\\ \2// / |||1 + tan |- + --|| | \ \\\ \2 4 // // \\\ \2 4 // /
$$\left(\left(\begin{cases} 1 & \text{for}\: \left(t + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}} & \text{otherwise} \end{cases}\right) - 1\right) \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} - 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(t + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)$$
(-1 + 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))*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))