Господин Экзамен
Другие калькуляторы
-
Как пользоваться?
- sin(t)^2+2*cos(t)^2-1 если t=1/3
- 25*sin(x)^2*cos(y)^2+25*sin(x)^2*sin(y)^2+16*cos(x)^2 если y=1/3
- sqrt(a*sqrt(a*sqrt(a*sqrt(a)))) если a=1/2
- 16-6*sin(x)^2-6*cos(x)^2 если x=-3/2
- Разложить многочлен на множители:
- 16*m^6*n^2-81*x^2
- x^2-5*x+4
- 7*m-n+49*m^2-n^2
- x^2-13*x+42
- Общий знаменатель:
- x-3/4*x^2+24*x+36/(x/3*x-9-3/x^2+3*x+x^2+9/27-3*x^2)
- cos(2*pi)/3-cos(4*pi)/3+sin(3*pi/2)*sin(5*pi)/6
- (1/x-y-1/x+y)/y/x-y
- tan(pi+a)*sin(pi/2-a)/cos(3*pi/2+a)
- Умножение столбиком:
- 16*38
- 22*45
- 815*204
- 25*50
- Деление столбиком:
- 7236/6
- 279/3
- 274/5
- 82/4
- Раскрыть скобки в:
- (c+b)*(c-b)-(5*c^2-b^2)
- 6*(m-n)
- 6*b-7*(12-3*b)
- (12-d)*(12+d)
- Разложение числа на простые множители:
- 20736
- 1260
- 7688
- 26
- Производная:
- 1/3
- График функции y =:
- 1/3
- Интеграл d{x}:
-
1/3
-
Идентичные выражения
- sin(t)^ два + два *cos(t)^ два - один если t= один / три
- синус от (t) в квадрате плюс 2 умножить на косинус от (t) в квадрате минус 1 если t равно 1 делить на 3
- синус от (t) в степени два плюс два умножить на косинус от (t) в степени два минус один если t равно один делить на три
- sin(t)2+2*cos(t)2-1 если t=1/3
- sint2+2*cost2-1 если t=1/3
- sin(t)²+2*cos(t)²-1 если t=1/3
- sin(t) в степени 2+2*cos(t) в степени 2-1 если t=1/3
- sin(t)^2+2cos(t)^2-1 если t=1/3
- sin(t)2+2cos(t)2-1 если t=1/3
- sint2+2cost2-1 если t=1/3
- sint^2+2cost^2-1 если t=1/3
- sin(t)^2+2*cos(t)^2-1 при t=1/3
- sin(t)^2+2*cos(t)^2-1 если t=1 разделить на 3
-
Похожие выражения
- sin(t)^2-2*cos(t)^2-1 если t=1/3
- 21/(3*a-a^2)-7/a если a=-3
- sin(t)^2+2*cos(t)^2+1 если t=1/3
sin(t)^2+2*cos(t)^2-1 если t=1/3
Решение
Вы ввели
[src]
2 2 sin (t) + 2*cos (t) - 1
$$\sin^{2}{\left(t \right)} + 2 \cos^{2}{\left(t \right)} - 1$$
sin(t)^2 + 2*cos(t)^2 - 1*1
Подстановка условия
[src]
sin(t)^2 + 2*cos(t)^2 - 1*1 при t = 1/3
подставляем
2 2 sin (t) + 2*cos (t) - 1
$$\sin^{2}{\left(t \right)} + 2 \cos^{2}{\left(t \right)} - 1$$
2 cos (t)
$$\cos^{2}{\left(t \right)}$$
переменные
t = 1/3
$$t = \frac{1}{3}$$
2 cos ((1/3))
$$\cos^{2}{\left((1/3) \right)}$$
2 cos (1/3)
$$\cos^{2}{\left(\frac{1}{3} \right)}$$
cos(1/3)^2
Степени
[src]
2 2
/ I*t -I*t\ / -I*t I*t\
|e e | \- e + e /
-1 + 2*|---- + -----| - -----------------
\ 2 2 / 4
$$2 \left(\frac{e^{i t}}{2} + \frac{e^{- i t}}{2}\right)^{2} - \frac{\left(e^{i t} - e^{- i t}\right)^{2}}{4} - 1$$
-1 + 2*(exp(i*t)/2 + exp(-i*t)/2)^2 - (-exp(-i*t) + exp(i*t))^2/4
Собрать выражение
[src]
1 cos(2*t) - + -------- 2 2
$$\frac{\cos{\left(2 t \right)}}{2} + \frac{1}{2}$$
1/2 + cos(2*t)/2
Тригонометрическая часть
[src]
2 cos (t)
$$\cos^{2}{\left(t \right)}$$
1 ------- 2 sec (t)
$$\frac{1}{\sec^{2}{\left(t \right)}}$$
2/ pi\
sin |t + --|
\ 2 /
$$\sin^{2}{\left(t + \frac{\pi}{2} \right)}$$
1 cos(2*t) - + -------- 2 2
$$\frac{\cos{\left(2 t \right)}}{2} + \frac{1}{2}$$
1
------------
2/pi \
csc |-- - t|
\2 /
$$\frac{1}{\csc^{2}{\left(- t + \frac{\pi}{2} \right)}}$$
2 2/ pi\
-1 + sin (t) + 2*sin |t + --|
\ 2 /
$$\sin^{2}{\left(t \right)} + 2 \sin^{2}{\left(t + \frac{\pi}{2} \right)} - 1$$
2/ pi\ 2
-1 + cos |t - --| + 2*cos (t)
\ 2 /
$$2 \cos^{2}{\left(t \right)} + \cos^{2}{\left(t - \frac{\pi}{2} \right)} - 1$$
1 2
-1 + ------- + -------
2 2
csc (t) sec (t)
$$-1 + \frac{2}{\sec^{2}{\left(t \right)}} + \frac{1}{\csc^{2}{\left(t \right)}}$$
2 2 1 cos (t) sin (t) - + ------- - ------- 2 2 2
$$- \frac{\sin^{2}{\left(t \right)}}{2} + \frac{\cos^{2}{\left(t \right)}}{2} + \frac{1}{2}$$
1 2
-1 + ------------ + -------
2/ pi\ 2
sec |t - --| sec (t)
\ 2 /
$$-1 + \frac{1}{\sec^{2}{\left(t - \frac{\pi}{2} \right)}} + \frac{2}{\sec^{2}{\left(t \right)}}$$
1 2
-1 + ------------ + -------
2/pi \ 2
sec |-- - t| sec (t)
\2 /
$$-1 + \frac{1}{\sec^{2}{\left(- t + \frac{\pi}{2} \right)}} + \frac{2}{\sec^{2}{\left(t \right)}}$$
1 2
-1 + ------- + ------------
2 2/pi \
csc (t) csc |-- - t|
\2 /
$$-1 + \frac{2}{\csc^{2}{\left(- t + \frac{\pi}{2} \right)}} + \frac{1}{\csc^{2}{\left(t \right)}}$$
2 2 - (1 - cos(t)) - 2*cos(t) + cos(2*t)
$$- \left(- \cos{\left(t \right)} + 1\right)^{2} - 2 \cos{\left(t \right)} + \cos{\left(2 t \right)} + 2$$
1 2
-1 + ------------ + ------------
2 2/pi \
csc (pi - t) csc |-- - t|
\2 /
$$-1 + \frac{2}{\csc^{2}{\left(- t + \frac{\pi}{2} \right)}} + \frac{1}{\csc^{2}{\left(- t + \pi \right)}}$$
/ 1 for t mod 2*pi = 0 | < 2 |cos (t) otherwise \
$$\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\cos^{2}{\left(t \right)} & \text{otherwise} \end{cases}$$
2
/ 2/t\\
|1 - tan |-||
\ \2//
--------------
2
/ 2/t\\
|1 + tan |-||
\ \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 pi\\ 2
|1 - cot |- + --|| *(1 + sin(t))
\ \2 4 //
--------------------------------- + cos(2*t)
4
$$\frac{\left(- \cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\sin{\left(t \right)} + 1\right)^{2}}{4} + \cos{\left(2 t \right)}$$
2/t pi\
8*tan |- + --|
1 - cos(2*t) \2 4 /
-1 + ------------ + -------------------
2 2
/ 2/t pi\\
|1 + tan |- + --||
\ \2 4 //
$$\frac{- \cos{\left(2 t \right)} + 1}{2} - 1 + \frac{8 \tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
/ 1 for t mod 2*pi = 0 | | 2 |/ 2/t\\ ||-1 + cot |-|| <\ \2// |--------------- otherwise | 2 | / 2/t\\ | |1 + cot |-|| \ \ \2//
$$\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{t}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}$$
2
/ 2/t\\ 2/t\
2*|1 - tan |-|| 4*tan |-|
\ \2// \2/
-1 + ---------------- + --------------
2 2
/ 2/t\\ / 2/t\\
|1 + tan |-|| |1 + tan |-||
\ \2// \ \2//
$$\frac{2 \left(- \tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} - 1 + \frac{4 \tan^{2}{\left(\frac{t}{2} \right)}}{\left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}}$$
2/t\ 2/t pi\
4*cot |-| 8*tan |- + --|
\2/ \2 4 /
-1 + -------------- + -------------------
2 2
/ 2/t\\ / 2/t pi\\
|1 + cot |-|| |1 + tan |- + --||
\ \2// \ \2 4 //
$$-1 + \frac{4 \cot^{2}{\left(\frac{t}{2} \right)}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} + \frac{8 \tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
2/t\ 2/t pi\
4*tan |-| 8*tan |- + --|
\2/ \2 4 /
-1 + -------------- + -------------------
2 2
/ 2/t\\ / 2/t pi\\
|1 + tan |-|| |1 + tan |- + --||
\ \2// \ \2 4 //
$$-1 + \frac{8 \tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} + \frac{4 \tan^{2}{\left(\frac{t}{2} \right)}}{\left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}}$$
2 2
(1 - cos(t) + sin(t)) *(-1 + cos(t) + sin(t))
----------------------------------------------
2
/1 2 cos(2*t)\
|- + (1 - cos(t)) - --------|
\2 2 /
$$\frac{\left(\sin{\left(t \right)} - \cos{\left(t \right)} + 1\right)^{2} \left(\sin{\left(t \right)} + \cos{\left(t \right)} - 1\right)^{2}}{\left(\left(- \cos{\left(t \right)} + 1\right)^{2} - \frac{\cos{\left(2 t \right)}}{2} + \frac{1}{2}\right)^{2}}$$
// 1 for t mod 2*pi = 0\ // 0 for t mod pi = 0\
|| | || |
-1 + 2*|< 2 | + |< 2 |
||cos (t) otherwise | ||sin (t) otherwise |
\\ / \\ /
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\sin^{2}{\left(t \right)} & \text{otherwise} \end{cases}\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\cos^{2}{\left(t \right)} & \text{otherwise} \end{cases}\right)\right) - 1$$
2
/ 1 \
2*|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//
$$\frac{2 \left(1 - \frac{1}{\cot^{2}{\left(\frac{t}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{t}{2} \right)}}\right)^{2}} - 1 + \frac{4}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \cot^{2}{\left(\frac{t}{2} \right)}}$$
// 1 for t mod 2*pi = 0\ // 0 for t mod pi = 0\
|| | || |
-1 + 2*|< 2 | + |< 2/ pi\ |
||cos (t) otherwise | ||cos |t - --| otherwise |
\\ / \\ \ 2 / /
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\cos^{2}{\left(t - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\cos^{2}{\left(t \right)} & \text{otherwise} \end{cases}\right)\right) - 1$$
// 1 for t mod 2*pi = 0\ // 0 for t mod pi = 0\
|| | || |
-1 + 2*|< 2/ pi\ | + |< 2 |
||sin |t + --| otherwise | ||sin (t) otherwise |
\\ \ 2 / / \\ /
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\sin^{2}{\left(t \right)} & \text{otherwise} \end{cases}\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\sin^{2}{\left(t + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) - 1$$
2 2
/ 2/t pi\\ / 2/t\\
|-1 + tan |- + --|| 2*|-1 + cot |-||
\ \2 4 // \ \2//
-1 + -------------------- + -----------------
2 2
/ 2/t pi\\ / 2/t\\
|1 + tan |- + --|| |1 + cot |-||
\ \2 4 // \ \2//
$$\frac{\left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} + \frac{2 \left(\cot^{2}{\left(\frac{t}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} - 1$$
2 2
/ 2/t pi\\ / 2/t\\
|1 - cot |- + --|| 2*|1 - tan |-||
\ \2 4 // \ \2//
-1 + ------------------- + ----------------
2 2
/ 2/t pi\\ / 2/t\\
|1 + cot |- + --|| |1 + tan |-||
\ \2 4 // \ \2//
$$\frac{2 \left(- \tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} + \frac{\left(- \cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}{\left(\cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} - 1$$
// 1 for t mod 2*pi = 0\ // 0 for t mod pi = 0\
|| | || |
|| 1 | || 1 |
-1 + 2*|<------- otherwise | + |<------------ otherwise |
|| 2 | || 2/ pi\ |
||sec (t) | ||sec |t - --| |
\\ / \\ \ 2 / /
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{1}{\sec^{2}{\left(t - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{1}{\sec^{2}{\left(t \right)}} & \text{otherwise} \end{cases}\right)\right) - 1$$
// 1 for t mod 2*pi = 0\ // 0 for t mod pi = 0\
|| | || |
|| 1 | || 1 |
-1 + 2*|<------------ otherwise | + |<------- otherwise |
|| 2/pi \ | || 2 |
||csc |-- - t| | ||csc (t) |
\\ \2 / / \\ /
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{1}{\csc^{2}{\left(t \right)}} & \text{otherwise} \end{cases}\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{1}{\csc^{2}{\left(- t + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - 1$$
// / 3*pi\ \
// 1 for t mod 2*pi = 0\ || 1 for |t + ----| mod 2*pi = 0|
|| | || \ 2 / |
-1 + 2*|< 2 | + |< |
||cos (t) otherwise | || 4/t\ 2/t\ |
\\ / ||- 4*cos |-| + 4*cos |-| otherwise |
\\ \2/ \2/ /
$$\left(2 \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\cos^{2}{\left(t \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \left(t + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\- 4 \cos^{4}{\left(\frac{t}{2} \right)} + 4 \cos^{2}{\left(\frac{t}{2} \right)} & \text{otherwise} \end{cases}\right) - 1$$
// / pi\ \
|| 0 for |t + --| mod pi = 0| // 0 for t mod pi = 0\
|| \ 2 / | || |
-1 + 2*|< | + |< 2 |
|| 2 2/t pi\ | ||sin (t) otherwise |
||(1 + sin(t)) *cot |- + --| otherwise | \\ /
\\ \2 4 / /
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\sin^{2}{\left(t \right)} & \text{otherwise} \end{cases}\right) + \left(2 \left(\begin{cases} 0 & \text{for}\: \left(t + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(t \right)} + 1\right)^{2} \cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)\right) - 1$$
2
/ 4/t\\
| 4*sin |-||
| \2/|
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) /
$$\frac{2 \left(- \frac{4 \sin^{4}{\left(\frac{t}{2} \right)}}{\sin^{2}{\left(t \right)}} + 1\right)^{2}}{\left(\frac{4 \sin^{4}{\left(\frac{t}{2} \right)}}{\sin^{2}{\left(t \right)}} + 1\right)^{2}} - 1 + \frac{16 \sin^{4}{\left(\frac{t}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{t}{2} \right)}}{\sin^{2}{\left(t \right)}} + 1\right)^{2} \sin^{2}{\left(t \right)}}$$
// 1 for t mod 2*pi = 0\ // 0 for t mod pi = 0\
|| | || |
||/ 1 for t mod 2*pi = 0 | ||/ 0 for t mod pi = 0 |
-1 + 2*|<| | + |<| |
||< 2 otherwise | ||< 2 otherwise |
|||cos (t) otherwise | |||sin (t) otherwise |
\\\ / \\\ /
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\sin^{2}{\left(t \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\cos^{2}{\left(t \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) - 1$$
// 1 for t mod 2*pi = 0\ // 0 for t mod pi = 0\
|| | || |
|| 2 | || 2/t\ |
||/ 2/t\\ | || 4*cot |-| |
|||-1 + cot |-|| | || \2/ |
-1 + 2*|<\ \2// | + |<-------------- otherwise |
||--------------- otherwise | || 2 |
|| 2 | ||/ 2/t\\ |
|| / 2/t\\ | |||1 + cot |-|| |
|| |1 + cot |-|| | ||\ \2// |
\\ \ \2// / \\ /
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{t}{2} \right)}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{t}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) - 1$$
// 1 for t mod 2*pi = 0\ // 0 for t mod pi = 0\
|| | || |
|| 2 | || 2/t\ |
||/ 2/t\\ | || 4*tan |-| |
|||1 - tan |-|| | || \2/ |
-1 + 2*|<\ \2// | + |<-------------- otherwise |
||-------------- otherwise | || 2 |
|| 2 | ||/ 2/t\\ |
||/ 2/t\\ | |||1 + tan |-|| |
|||1 + tan |-|| | ||\ \2// |
\\\ \2// / \\ /
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{4 \tan^{2}{\left(\frac{t}{2} \right)}}{\left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(- \tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) - 1$$
// 1 for t mod 2*pi = 0\
|| |
|| 2 | // 0 for t mod pi = 0\
||/ 1 \ | || |
|||-1 + -------| | || 4 |
||| 2/t\| | ||---------------------- otherwise |
||| tan |-|| | || 2 |
-1 + 2*|<\ \2// | + |
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{4}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \tan^{2}{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{t}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{t}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)\right) - 1$$
2
/ 2/t pi\\
| cos |- - --||
| \2 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/ /
$$\frac{2 \left(1 - \frac{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} \right)}}\right)^{2}}{\left(1 + \frac{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} \right)}}\right)^{2}} - 1 + \frac{4 \cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \cos^{2}{\left(\frac{t}{2} \right)}}$$
2
/ 2/t\ \
| sec |-| |
| \2/ |
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 //
$$\frac{2 \left(- \frac{\sec^{2}{\left(\frac{t}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}}{\left(\frac{\sec^{2}{\left(\frac{t}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}} - 1 + \frac{4 \sec^{2}{\left(\frac{t}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{t}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}$$
2
/ 2/pi t\\
| csc |-- - -||
| \2 2/|
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/ /
$$\frac{2 \left(1 - \frac{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{t}{2} \right)}}\right)^{2}}{\left(1 + \frac{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{t}{2} \right)}}\right)^{2}} - 1 + \frac{4 \csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \csc^{2}{\left(\frac{t}{2} \right)}}$$
// / pi\ \
|| 0 for |t + --| mod pi = 0| // 0 for t mod pi = 0\
|| \ 2 / | || |
|| | || 2/t\ |
|| 2/t pi\ | || 4*cot |-| |
|| 4*cot |- + --| | || \2/ |
-1 + 2*|< \2 4 / | + |<-------------- otherwise |
||------------------- otherwise | || 2 |
|| 2 | ||/ 2/t\\ |
||/ 2/t pi\\ | |||1 + cot |-|| |
|||1 + cot |- + --|| | ||\ \2// |
||\ \2 4 // | \\ /
\\ /
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{t}{2} \right)}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) + \left(2 \left(\begin{cases} 0 & \text{for}\: \left(t + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)}}{\left(\cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) - 1$$
// / 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*|<\ \2// | + |<|-1 + tan |- + --|| |
||--------------- otherwise | ||\ \2 4 // |
|| 2 | ||-------------------- otherwise |
|| / 2/t\\ | || 2 |
|| |1 + cot |-|| | ||/ 2/t pi\\ |
\\ \ \2// / |||1 + tan |- + --|| |
\\\ \2 4 // /
$$\left(2 \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{t}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \left(t + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) - 1$$
// 0 for t mod pi = 0\
// 1 for t mod 2*pi = 0\ || |
|| | || 2 |
|| 2 | || sin (t) |
||/ 2 4/t\\ | ||------------------------ otherwise |
|||sin (t) - 4*sin |-|| | || 2 |
-1 + 2*|<\ \2// | + |
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{\sin^{2}{\left(t \right)}}{\left(1 + \frac{\sin^{2}{\left(t \right)}}{4 \sin^{4}{\left(\frac{t}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(- 4 \sin^{4}{\left(\frac{t}{2} \right)} + \sin^{2}{\left(t \right)}\right)^{2}}{\left(4 \sin^{4}{\left(\frac{t}{2} \right)} + \sin^{2}{\left(t \right)}\right)^{2}} & \text{otherwise} \end{cases}\right)\right) - 1$$
// 1 for t mod 2*pi = 0\
|| |
|| 2 | // 0 for t mod pi = 0\
||/ 2 \ | || |
||| sin (t) | | || 2 |
|||-1 + ---------| | || sin (t) |
||| 4/t\| | ||------------------------ otherwise |
||| 4*sin |-|| | || 2 |
-1 + 2*|<\ \2// | + |
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{\sin^{2}{\left(t \right)}}{\left(1 + \frac{\sin^{2}{\left(t \right)}}{4 \sin^{4}{\left(\frac{t}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sin^{2}{\left(t \right)}}{4 \sin^{4}{\left(\frac{t}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sin^{2}{\left(t \right)}}{4 \sin^{4}{\left(\frac{t}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)\right) - 1$$
// 1 for t mod 2*pi = 0\ // 0 for t mod pi = 0\
|| | || |
||/ 1 for t mod 2*pi = 0 | ||/ 0 for t mod pi = 0 |
||| | ||| |
||| 2 | ||| 2/t\ |
|||/ 2/t\\ | ||| 4*cot |-| |
-1 + 2*|<||-1 + cot |-|| | + |<| \2/ |
||<\ \2// otherwise | ||<-------------- otherwise otherwise |
|||--------------- otherwise | ||| 2 |
||| 2 | |||/ 2/t\\ |
||| / 2/t\\ | ||||1 + cot |-|| |
||| |1 + cot |-|| | |||\ \2// |
\\\ \ \2// / \\\ /
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{t}{2} \right)}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{t}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) - 1$$
// 1 for t mod 2*pi = 0\
|| |
|| 2 | // 0 for t mod pi = 0\
||/ 2/t\ \ | || |
||| cos |-| | | || 2/t\ |
||| \2/ | | || 4*cos |-| |
|||-1 + ------------| | || \2/ |
||| 2/t pi\| | ||-------------------------------- otherwise |
||| cos |- - --|| | || 2 |
-1 + 2*|<\ \2 2 // | + |
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{4 \cos^{2}{\left(\frac{t}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{t}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(\frac{\cos^{2}{\left(\frac{t}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\cos^{2}{\left(\frac{t}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) - 1$$
// 1 for t mod 2*pi = 0\
|| |
|| 2 | // 0 for t mod pi = 0\
||/ 2/t pi\\ | || |
||| sec |- - --|| | || 2/t pi\ |
||| \2 2 /| | || 4*sec |- - --| |
|||-1 + ------------| | || \2 2 / |
||| 2/t\ | | ||--------------------------- otherwise |
||| sec |-| | | || 2 |
-1 + 2*|<\ \2/ / | + |
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{4 \sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \sec^{2}{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)\right) - 1$$
// 1 for t mod 2*pi = 0\
|| |
|| 2 | // 0 for t mod pi = 0\
||/ 2/t\ \ | || |
||| csc |-| | | || 2/t\ |
||| \2/ | | || 4*csc |-| |
|||-1 + ------------| | || \2/ |
||| 2/pi t\| | ||-------------------------------- otherwise |
||| csc |-- - -|| | || 2 |
-1 + 2*|<\ \2 2// | + |
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{4 \csc^{2}{\left(\frac{t}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{t}{2} \right)}}{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2} \csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{t}{2} \right)}}{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\csc^{2}{\left(\frac{t}{2} \right)}}{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) - 1$$
-1 + 2*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)) + 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))