Господин Экзамен
Другие калькуляторы
-
Как пользоваться?
- 1-cos(t)^2/1-sin(t)^2 если t=-4
- (4*x^2-40*x*y+100*y^2)/(15*y-3*x) если y=2
- a+3 если a=-2
- a*4*sqrt(a^6) если a=-4
- Разложить многочлен на множители:
- x^2-15
- Y^5-32
- 0.09-a^2
- 8+x^3
- Общий знаменатель:
- (a-b)*2+a*b/(a+b)*2-a*b/a5+a3*b2+a2*b3+b5/(a3+a2*b+b2*a+b3)*(a3-b3)
- (((d-9)/(d))-((c-9)/(c)))*((c*d)/(d^2-c^2))
- 2*cos(-2*pi-b)+sin(-pi/2+b)/2*cos(b+pi)
- pi/2-pi/4
- Умножение столбиком:
- 773*797
- 36*25
- 90*15
- 70*13
- Деление столбиком:
- 65376/2
- 876/7
- 12240/3
- 70/5
- Раскрыть скобки в:
- (2*k-3)*(3+2*k)-(3*k-5)^2-30*k
- (x-7)*(x+1)
- (5-x)*(x-5)
- (a^(-5))^2*a^12
- Разложение числа на простые множители:
- 388
- 4860
- 6652
- 23716
- Производная:
- -4
- Интеграл d{x}:
-
-4
- График функции y =:
- -4
-
Идентичные выражения
- один -cos(t)^ два / один -sin(t)^ два если t=- четыре
- 1 минус косинус от (t) в квадрате делить на 1 минус синус от (t) в квадрате если t равно минус 4
- один минус косинус от (t) в степени два делить на один минус синус от (t) в степени два если t равно минус четыре
- 1-cos(t)2/1-sin(t)2 если t=-4
- 1-cost2/1-sint2 если t=-4
- 1-cos(t)²/1-sin(t)² если t=-4
- 1-cos(t) в степени 2/1-sin(t) в степени 2 если t=-4
- 1-cost^2/1-sint^2 если t=-4
- 1-cos(t)^2/1-sin(t)^2 при t=-4
- 1-cos(t)^2 разделить на 1-sin(t)^2 если t=-4
-
Похожие выражения
- 1-cos(t)^2/1-sin(t)^2 если t=+4
- 1+cos(t)^2/1-sin(t)^2 если t=-4
- 1-cos(t)^2/1+sin(t)^2 если t=-4
1-cos(t)^2/1-sin(t)^2 если t=-4
Решение
Вы ввели
[src]
2
cos (t) 2
1 - ------- - sin (t)
1
$$- \sin^{2}{\left(t \right)} - \frac{\cos^{2}{\left(t \right)}}{1} + 1$$
1 - cos(t)^2/1 - sin(t)^2
Комбинаторика
[src]
2 2 1 - cos (t) - sin (t)
$$- \sin^{2}{\left(t \right)} - \cos^{2}{\left(t \right)} + 1$$
1 - cos(t)^2 - sin(t)^2
Объединение рациональных выражений
[src]
2 2 1 - cos (t) - sin (t)
$$- \sin^{2}{\left(t \right)} - \cos^{2}{\left(t \right)} + 1$$
1 - cos(t)^2 - sin(t)^2
Общий знаменатель
[src]
2 2 1 - cos (t) - sin (t)
$$- \sin^{2}{\left(t \right)} - \cos^{2}{\left(t \right)} + 1$$
1 - cos(t)^2 - sin(t)^2
Рациональный знаменатель
[src]
2 2 1 - cos (t) - sin (t)
$$- \sin^{2}{\left(t \right)} - \cos^{2}{\left(t \right)} + 1$$
1 - cos(t)^2 - sin(t)^2
Степени
[src]
2 2 1 - cos (t) - sin (t)
$$- \sin^{2}{\left(t \right)} - \cos^{2}{\left(t \right)} + 1$$
2 2
/ I*t -I*t\ / -I*t I*t\
|e e | \- e + e /
1 - |---- + -----| + -----------------
\ 2 2 / 4
$$- \left(\frac{e^{i t}}{2} + \frac{e^{- i t}}{2}\right)^{2} + \frac{\left(e^{i t} - e^{- i t}\right)^{2}}{4} + 1$$
1 - (exp(i*t)/2 + exp(-i*t)/2)^2 + (-exp(-i*t) + exp(i*t))^2/4
Собрать выражение
[src]
0
$$0$$
2 2 1 - cos (t) - sin (t)
$$- \sin^{2}{\left(t \right)} - \cos^{2}{\left(t \right)} + 1$$
1 - cos(t)^2 - sin(t)^2
Тригонометрическая часть
[src]
0
$$0$$
2 2 1 - cos (t) - sin (t)
$$- \sin^{2}{\left(t \right)} - \cos^{2}{\left(t \right)} + 1$$
2 2/ pi\
1 - cos (t) - cos |t - --|
\ 2 /
$$- \cos^{2}{\left(t \right)} - \cos^{2}{\left(t - \frac{\pi}{2} \right)} + 1$$
2 2/ pi\
1 - sin (t) - sin |t + --|
\ 2 /
$$- \sin^{2}{\left(t \right)} - \sin^{2}{\left(t + \frac{\pi}{2} \right)} + 1$$
1 1
1 - ------- - -------
2 2
csc (t) sec (t)
$$1 - \frac{1}{\sec^{2}{\left(t \right)}} - \frac{1}{\csc^{2}{\left(t \right)}}$$
1 1
1 - ------- - ------------
2 2/ pi\
sec (t) sec |t - --|
\ 2 /
$$1 - \frac{1}{\sec^{2}{\left(t - \frac{\pi}{2} \right)}} - \frac{1}{\sec^{2}{\left(t \right)}}$$
1 1
1 - ------- - ------------
2 2/pi \
sec (t) sec |-- - t|
\2 /
$$1 - \frac{1}{\sec^{2}{\left(- t + \frac{\pi}{2} \right)}} - \frac{1}{\sec^{2}{\left(t \right)}}$$
1 1
1 - ------- - ------------
2 2/pi \
csc (t) csc |-- - t|
\2 /
$$1 - \frac{1}{\csc^{2}{\left(- t + \frac{\pi}{2} \right)}} - \frac{1}{\csc^{2}{\left(t \right)}}$$
1 1
1 - ------------ - ------------
2 2/pi \
csc (pi - t) csc |-- - t|
\2 /
$$1 - \frac{1}{\csc^{2}{\left(- t + \frac{\pi}{2} \right)}} - \frac{1}{\csc^{2}{\left(- t + \pi \right)}}$$
3 2 cos(2*t) - - + (1 - cos(t)) + 2*cos(t) - -------- 2 2
$$\left(- \cos{\left(t \right)} + 1\right)^{2} + 2 \cos{\left(t \right)} - \frac{\cos{\left(2 t \right)}}{2} - \frac{3}{2}$$
2
/ 2/t pi\\ 2
|1 - cot |- + --|| *(1 + sin(t))
1 + cos(2*t) \ \2 4 //
1 - ------------ - ---------------------------------
2 4
$$- \frac{\left(- \cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\sin{\left(t \right)} + 1\right)^{2}}{4} - \frac{\cos{\left(2 t \right)} + 1}{2} + 1$$
2/t pi\
4*tan |- + --|
1 - cos(2*t) \2 4 /
1 - ------------ - -------------------
2 2
/ 2/t pi\\
|1 + tan |- + --||
\ \2 4 //
$$- \frac{- \cos{\left(2 t \right)} + 1}{2} + 1 - \frac{4 \tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
2
/ 2/t\\ 2/t\
|1 - tan |-|| 4*tan |-|
\ \2// \2/
1 - -------------- - --------------
2 2
/ 2/t\\ / 2/t\\
|1 + tan |-|| |1 + tan |-||
\ \2// \ \2//
$$- \frac{\left(- \tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} + 1 - \frac{4 \tan^{2}{\left(\frac{t}{2} \right)}}{\left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}}$$
2/t\ 2/t pi\
4*cot |-| 4*tan |- + --|
\2/ \2 4 /
1 - -------------- - -------------------
2 2
/ 2/t\\ / 2/t pi\\
|1 + cot |-|| |1 + tan |- + --||
\ \2// \ \2 4 //
$$1 - \frac{4 \cot^{2}{\left(\frac{t}{2} \right)}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} - \frac{4 \tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
2/t\ 2/t pi\
4*tan |-| 4*tan |- + --|
\2/ \2 4 /
1 - -------------- - -------------------
2 2
/ 2/t\\ / 2/t pi\\
|1 + tan |-|| |1 + tan |- + --||
\ \2// \ \2 4 //
$$1 - \frac{4 \tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} - \frac{4 \tan^{2}{\left(\frac{t}{2} \right)}}{\left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}}$$
// 0 for t mod pi = 0\ // 1 for t mod 2*pi = 0\
|| | || |
1 - |< 2 | - |< 2 |
||sin (t) otherwise | ||cos (t) otherwise |
\\ / \\ /
$$\left(- \begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\sin^{2}{\left(t \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\cos^{2}{\left(t \right)} & \text{otherwise} \end{cases}\right) + 1$$
2
/ 1 \
|1 - -------|
| 2/t\|
| cot |-||
\ \2// 4
1 - -------------- - ----------------------
2 2
/ 1 \ / 1 \ 2/t\
|1 + -------| |1 + -------| *cot |-|
| 2/t\| | 2/t\| \2/
| cot |-|| | cot |-||
\ \2// \ \2//
$$- \frac{\left(1 - \frac{1}{\cot^{2}{\left(\frac{t}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{t}{2} \right)}}\right)^{2}} + 1 - \frac{4}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \cot^{2}{\left(\frac{t}{2} \right)}}$$
// 0 for t mod pi = 0\ // 1 for t mod 2*pi = 0\
|| | || |
1 - |< 2 | - |< 2/ pi\ |
||sin (t) otherwise | ||sin |t + --| otherwise |
\\ / \\ \ 2 / /
$$\left(- \begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\sin^{2}{\left(t \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\sin^{2}{\left(t + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + 1$$
// 0 for t mod pi = 0\ // 1 for t mod 2*pi = 0\
|| | || |
1 - |< 2/ pi\ | - |< 2 |
||cos |t - --| otherwise | ||cos (t) otherwise |
\\ \ 2 / / \\ /
$$\left(- \begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\cos^{2}{\left(t - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\cos^{2}{\left(t \right)} & \text{otherwise} \end{cases}\right) + 1$$
// 0 for t mod pi = 0\ // 1 for t mod 2*pi = 0\
|| | || |
|| 1 | || 1 |
1 - |<------------ otherwise | - |<------- otherwise |
|| 2/ pi\ | || 2 |
||sec |t - --| | ||sec (t) |
\\ \ 2 / / \\ /
$$\left(- \begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{1}{\sec^{2}{\left(t - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{1}{\sec^{2}{\left(t \right)}} & \text{otherwise} \end{cases}\right) + 1$$
2 2
/ 2/t\\ / 2/t pi\\
|-1 + cot |-|| |-1 + tan |- + --||
\ \2// \ \2 4 //
1 - --------------- - --------------------
2 2
/ 2/t\\ / 2/t pi\\
|1 + cot |-|| |1 + tan |- + --||
\ \2// \ \2 4 //
$$- \frac{\left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} - \frac{\left(\cot^{2}{\left(\frac{t}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} + 1$$
// 0 for t mod pi = 0\ // 1 for t mod 2*pi = 0\
|| | || |
|| 1 | || 1 |
1 - |<------- otherwise | - |<------------ otherwise |
|| 2 | || 2/pi \ |
||csc (t) | ||csc |-- - t| |
\\ / \\ \2 / /
$$\left(- \begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{1}{\csc^{2}{\left(t \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{1}{\csc^{2}{\left(- t + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1$$
2 2
/ 2/t pi\\ / 2/t\\
|1 - cot |- + --|| |1 - tan |-||
\ \2 4 // \ \2//
1 - ------------------- - --------------
2 2
/ 2/t pi\\ / 2/t\\
|1 + cot |- + --|| |1 + tan |-||
\ \2 4 // \ \2//
$$- \frac{\left(- \tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} - \frac{\left(- \cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}{\left(\cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} + 1$$
// / 3*pi\ \
// 1 for t mod 2*pi = 0\ || 1 for |t + ----| mod 2*pi = 0|
|| | || \ 2 / |
1 - |< 2 | - |< |
||cos (t) otherwise | || 4/t\ 2/t\ |
\\ / ||- 4*cos |-| + 4*cos |-| otherwise |
\\ \2/ \2/ /
$$\left(- \begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\cos^{2}{\left(t \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \left(t + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\- 4 \cos^{4}{\left(\frac{t}{2} \right)} + 4 \cos^{2}{\left(\frac{t}{2} \right)} & \text{otherwise} \end{cases}\right) + 1$$
// / pi\ \
// 0 for t mod pi = 0\ || 0 for |t + --| mod pi = 0|
|| | || \ 2 / |
1 - |< 2 | - |< |
||sin (t) otherwise | || 2 2/t pi\ |
\\ / ||(1 + sin(t)) *cot |- + --| otherwise |
\\ \2 4 / /
$$\left(- \begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\sin^{2}{\left(t \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 0 & \text{for}\: \left(t + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(t \right)} + 1\right)^{2} \cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right) + 1$$
2
/ 4/t\\
| 4*sin |-||
| \2/|
|1 - ---------| 4/t\
| 2 | 16*sin |-|
\ sin (t) / \2/
1 - ---------------- - ------------------------
2 2
/ 4/t\\ / 4/t\\
| 4*sin |-|| | 4*sin |-||
| \2/| | \2/| 2
|1 + ---------| |1 + ---------| *sin (t)
| 2 | | 2 |
\ sin (t) / \ sin (t) /
$$- \frac{\left(- \frac{4 \sin^{4}{\left(\frac{t}{2} \right)}}{\sin^{2}{\left(t \right)}} + 1\right)^{2}}{\left(\frac{4 \sin^{4}{\left(\frac{t}{2} \right)}}{\sin^{2}{\left(t \right)}} + 1\right)^{2}} + 1 - \frac{16 \sin^{4}{\left(\frac{t}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{t}{2} \right)}}{\sin^{2}{\left(t \right)}} + 1\right)^{2} \sin^{2}{\left(t \right)}}$$
// 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 |
1 - |<| | - |<| |
||< 2 otherwise | ||< 2 otherwise |
|||sin (t) otherwise | |||cos (t) otherwise |
\\\ / \\\ /
$$\left(- \begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\sin^{2}{\left(t \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\cos^{2}{\left(t \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + 1$$
// 0 for t mod pi = 0\ // 1 for t mod 2*pi = 0\
|| | || |
|| 2/t\ | || 2 |
|| 4*cot |-| | ||/ 2/t\\ |
|| \2/ | |||-1 + cot |-|| |
1 - |<-------------- otherwise | - |<\ \2// |
|| 2 | ||--------------- otherwise |
||/ 2/t\\ | || 2 |
|||1 + cot |-|| | || / 2/t\\ |
||\ \2// | || |1 + cot |-|| |
\\ / \\ \ \2// /
$$\left(- \begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{t}{2} \right)}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{t}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) + 1$$
// 0 for t mod pi = 0\ // 1 for t mod 2*pi = 0\
|| | || |
|| 2/t\ | || 2 |
|| 4*tan |-| | ||/ 2/t\\ |
|| \2/ | |||1 - tan |-|| |
1 - |<-------------- otherwise | - |<\ \2// |
|| 2 | ||-------------- otherwise |
||/ 2/t\\ | || 2 |
|||1 + tan |-|| | ||/ 2/t\\ |
||\ \2// | |||1 + tan |-|| |
\\ / \\\ \2// /
$$\left(- \begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{4 \tan^{2}{\left(\frac{t}{2} \right)}}{\left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(- \tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) + 1$$
// 1 for t mod 2*pi = 0\
|| |
// 0 for t mod pi = 0\ || 2 |
|| | ||/ 1 \ |
|| 4 | |||-1 + -------| |
||---------------------- otherwise | ||| 2/t\| |
|| 2 | ||| tan |-|| |
1 - |
$$\left(- \begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{4}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \tan^{2}{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{t}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{t}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right) + 1$$
// / pi\ \
// 0 for t mod pi = 0\ || 0 for |t + --| mod pi = 0|
|| | || \ 2 / |
|| 2/t\ | || |
|| 4*cot |-| | || 2/t pi\ |
|| \2/ | || 4*cot |- + --| |
1 - |<-------------- otherwise | - |< \2 4 / |
|| 2 | ||------------------- otherwise |
||/ 2/t\\ | || 2 |
|||1 + cot |-|| | ||/ 2/t pi\\ |
||\ \2// | |||1 + cot |- + --|| |
\\ / ||\ \2 4 // |
\\ /
$$\left(- \begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{t}{2} \right)}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 0 & \text{for}\: \left(t + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)}}{\left(\cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) + 1$$
2
/ 2/t pi\\
| cos |- - --||
| \2 2 /|
|1 - ------------|
| 2/t\ | 2/t pi\
| cos |-| | 4*cos |- - --|
\ \2/ / \2 2 /
1 - ------------------- - ---------------------------
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/ /
$$- \frac{\left(1 - \frac{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} \right)}}\right)^{2}}{\left(1 + \frac{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} \right)}}\right)^{2}} + 1 - \frac{4 \cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \cos^{2}{\left(\frac{t}{2} \right)}}$$
2
/ 2/t\ \
| sec |-| |
| \2/ |
|1 - ------------|
| 2/t pi\| 2/t\
| sec |- - --|| 4*sec |-|
\ \2 2 // \2/
1 - ------------------- - --------------------------------
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 //
$$- \frac{\left(- \frac{\sec^{2}{\left(\frac{t}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}}{\left(\frac{\sec^{2}{\left(\frac{t}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}} + 1 - \frac{4 \sec^{2}{\left(\frac{t}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{t}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}$$
2
/ 2/pi t\\
| csc |-- - -||
| \2 2/|
|1 - ------------|
| 2/t\ | 2/pi t\
| csc |-| | 4*csc |-- - -|
\ \2/ / \2 2/
1 - ------------------- - ---------------------------
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/ /
$$- \frac{\left(1 - \frac{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{t}{2} \right)}}\right)^{2}}{\left(1 + \frac{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{t}{2} \right)}}\right)^{2}} + 1 - \frac{4 \csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \csc^{2}{\left(\frac{t}{2} \right)}}$$
// / 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\\ |
1 - |<\ \2// | - |<|-1 + tan |- + --|| |
||--------------- otherwise | ||\ \2 4 // |
|| 2 | ||-------------------- otherwise |
|| / 2/t\\ | || 2 |
|| |1 + cot |-|| | ||/ 2/t pi\\ |
\\ \ \2// / |||1 + tan |- + --|| |
\\\ \2 4 // /
$$\left(- \begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{t}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \left(t + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) + 1$$
// 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 |-|| |
1 - |
$$\left(- \begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{\sin^{2}{\left(t \right)}}{\left(1 + \frac{\sin^{2}{\left(t \right)}}{4 \sin^{4}{\left(\frac{t}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(- 4 \sin^{4}{\left(\frac{t}{2} \right)} + \sin^{2}{\left(t \right)}\right)^{2}}{\left(4 \sin^{4}{\left(\frac{t}{2} \right)} + \sin^{2}{\left(t \right)}\right)^{2}} & \text{otherwise} \end{cases}\right) + 1$$
// 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 |-|| |
1 - |
$$\left(- \begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{\sin^{2}{\left(t \right)}}{\left(1 + \frac{\sin^{2}{\left(t \right)}}{4 \sin^{4}{\left(\frac{t}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sin^{2}{\left(t \right)}}{4 \sin^{4}{\left(\frac{t}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sin^{2}{\left(t \right)}}{4 \sin^{4}{\left(\frac{t}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right) + 1$$
// 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\\ |
1 - |<| \2/ | - |<||-1 + cot |-|| |
||<-------------- otherwise otherwise | ||<\ \2// otherwise |
||| 2 | |||--------------- otherwise |
|||/ 2/t\\ | ||| 2 |
||||1 + cot |-|| | ||| / 2/t\\ |
|||\ \2// | ||| |1 + cot |-|| |
\\\ / \\\ \ \2// /
$$\left(- \begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{t}{2} \right)}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{t}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + 1$$
// 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 |- - --|| |
1 - |
$$\left(- \begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{4 \cos^{2}{\left(\frac{t}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{t}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(\frac{\cos^{2}{\left(\frac{t}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\cos^{2}{\left(\frac{t}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) + 1$$
// 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 |-| | |
1 - |
$$\left(- \begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{4 \sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \sec^{2}{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right) + 1$$
// 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 |-- - -|| |
1 - |
$$\left(- \begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{4 \csc^{2}{\left(\frac{t}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{t}{2} \right)}}{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2} \csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{t}{2} \right)}}{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\csc^{2}{\left(\frac{t}{2} \right)}}{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) + 1$$
1 - 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))