Господин Экзамен
Другие калькуляторы
-
Как пользоваться?
- sin(t)^2-cos(t)^2/cot(-t)*tan(t) если t=-1/3
- 4^((sqrt(a-b))^4)-2*6^((sqrt(a+b))^6) если a=1/2
- b^(3*sqrt(2)+2)/((b^(sqrt(2)))^3) если b=3/2
- a^3*x^2-a*x-4*a^3-2*a если a=-3/2
- Разложить многочлен на множители:
- x^7-1
- 25-16*x^2
- x^6+y^6
- 49*k^2+42*k*p+9*p^2
- Общий знаменатель:
- sqrt(2)/(sqrt(8)+2)-((6-sqrt(32))/sqrt(8)-3)
- (x^2-9)/(x-6)^2-27/(6-x)^2
- 5/x-7-2/x-3+28/x^2-49
- b^2/x^4+b/x^2
- Умножение столбиком:
- 12*18
- 52*14
- 773*797
- 36*25
- Деление столбиком:
- 475/3
- 320/4
- 80/3
- 87/4
- Раскрыть скобки в:
- (6*c^5-7*d^2)*(6*c^5+7*d^2)
- (7-b)*(7-b)
- a*x*(x+a^2*x)
- (4*a^3+5)^2+(4*a^3-1)^2-2*(4*a^3+5)*(4*a^3-1)
- Разложение числа на простые множители:
- 31024
- 388
- 4860
- 6652
- Производная:
- -1/3
- Интеграл d{x}:
-
-1/3
-
Идентичные выражения
- sin(t)^ два -cos(t)^ два /cot(-t)*tan(t) если t=- один / три
- синус от (t) в квадрате минус косинус от (t) в квадрате делить на котангенс от ( минус t) умножить на тангенс от (t) если t равно минус 1 делить на 3
- синус от (t) в степени два минус косинус от (t) в степени два делить на котангенс от ( минус t) умножить на тангенс от (t) если t равно минус один делить на три
- sin(t)2-cos(t)2/cot(-t)*tan(t) если t=-1/3
- sint2-cost2/cot-t*tant если t=-1/3
- sin(t)²-cos(t)²/cot(-t)*tan(t) если t=-1/3
- sin(t) в степени 2-cos(t) в степени 2/cot(-t)*tan(t) если t=-1/3
- sin(t)^2-cos(t)^2/cot(-t)tan(t) если t=-1/3
- sin(t)2-cos(t)2/cot(-t)tan(t) если t=-1/3
- sint2-cost2/cot-ttant если t=-1/3
- sint^2-cost^2/cot-ttant если t=-1/3
- sin(t)^2-cos(t)^2/cot(-t)*tan(t) при t=-1/3
- sin(t)^2-cos(t)^2 разделить на cot(-t)*tan(t) если t=-1 разделить на 3
-
Похожие выражения
- sin(t)^2-cos(t)^2/cot(-t)*tan(t) если t=+1/3
- sin(t)^2+cos(t)^2/cot(-t)*tan(t) если t=-1/3
- sin(t)^2-cos(t)^2/cot(t)*tan(t) если t=-1/3
sin(t)^2-cos(t)^2/cot(-t)*tan(t) если t=-1/3
Решение
Вы ввели
[src]
2
2 cos (t)*tan(t)
sin (t) - --------------
cot(-t)
$$\sin^{2}{\left(t \right)} - \frac{\cos^{2}{\left(t \right)} \tan{\left(t \right)}}{\cot{\left(- t \right)}}$$
sin(t)^2 - cos(t)^2*tan(t)/cot(-t)
Общее упрощение
[src]
sin(2*t)*tan(t)
$$\sin{\left(2 t \right)} \tan{\left(t \right)}$$
sin(2*t)*tan(t)
Подстановка условия
[src]
sin(t)^2 - cos(t)^2*tan(t)/cot(-t) при t = -1/3
подставляем
2
2 cos (t)*tan(t)
sin (t) - --------------
cot(-t)
$$\sin^{2}{\left(t \right)} - \frac{\cos^{2}{\left(t \right)} \tan{\left(t \right)}}{\cot{\left(- t \right)}}$$
sin(2*t)*tan(t)
$$\sin{\left(2 t \right)} \tan{\left(t \right)}$$
переменные
t = -1/3
$$t = - \frac{1}{3}$$
sin(2*(-1/3))*tan((-1/3))
$$\sin{\left(2 (-1/3) \right)} \tan{\left((-1/3) \right)}$$
sin(2*-1/3)*tan(-1/3)
$$\sin{\left(2 \left(- \frac{1}{3}\right) \right)} \tan{\left(- \frac{1}{3} \right)}$$
sin(2/3)*tan(1/3)
$$\sin{\left(\frac{2}{3} \right)} \tan{\left(\frac{1}{3} \right)}$$
sin(2/3)*tan(1/3)
Объединение рациональных выражений
[src]
2 2
cos (t)*tan(t) + sin (t)*cot(t)
-------------------------------
cot(t)
$$\frac{\sin^{2}{\left(t \right)} \cot{\left(t \right)} + \cos^{2}{\left(t \right)} \tan{\left(t \right)}}{\cot{\left(t \right)}}$$
(cos(t)^2*tan(t) + sin(t)^2*cot(t))/cot(t)
Степени
[src]
2
2 cos (t)*tan(t)
sin (t) + --------------
cot(t)
$$\sin^{2}{\left(t \right)} + \frac{\cos^{2}{\left(t \right)} \tan{\left(t \right)}}{\cot{\left(t \right)}}$$
2
/ I*t -I*t\
2 |e e | / I*t -I*t\
/ -I*t I*t\ I*|---- + -----| *\- e + e /
\- e + e / \ 2 2 /
- ----------------- + ----------------------------------
4 / I*t -I*t\
\e + e /*cot(t)
$$- \frac{\left(e^{i t} - e^{- i t}\right)^{2}}{4} + \frac{i \left(- e^{i t} + e^{- i t}\right) \left(\frac{e^{i t}}{2} + \frac{e^{- i t}}{2}\right)^{2}}{\left(e^{i t} + e^{- i t}\right) \cot{\left(t \right)}}$$
-(-exp(-i*t) + exp(i*t))^2/4 + i*(exp(i*t)/2 + exp(-i*t)/2)^2*(-exp(i*t) + exp(-i*t))/((exp(i*t) + exp(-i*t))*cot(t))
Тригонометрическая часть
[src]
2 2*sin (t)
$$2 \sin^{2}{\left(t \right)}$$
1 - cos(2*t)
$$- \cos{\left(2 t \right)} + 1$$
2 ------- 2 csc (t)
$$\frac{2}{\csc^{2}{\left(t \right)}}$$
2 ------------ 2 csc (pi - t)
$$\frac{2}{\csc^{2}{\left(- t + \pi \right)}}$$
2/ pi\
2*cos |t - --|
\ 2 /
$$2 \cos^{2}{\left(t - \frac{\pi}{2} \right)}$$
2
------------
2/ pi\
sec |t - --|
\ 2 /
$$\frac{2}{\sec^{2}{\left(t - \frac{\pi}{2} \right)}}$$
2
------------
2/pi \
sec |-- - t|
\2 /
$$\frac{2}{\sec^{2}{\left(- t + \frac{\pi}{2} \right)}}$$
4/t\ 2/t\
- 8*cos |-| + 8*cos |-|
\2/ \2/
$$- 8 \cos^{4}{\left(\frac{t}{2} \right)} + 8 \cos^{2}{\left(\frac{t}{2} \right)}$$
2/t\
8*cot |-|
\2/
--------------
2
/ 2/t\\
|1 + cot |-||
\ \2//
$$\frac{8 \cot^{2}{\left(\frac{t}{2} \right)}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}}$$
2/t\
8*tan |-|
\2/
--------------
2
/ 2/t\\
|1 + tan |-||
\ \2//
$$\frac{8 \tan^{2}{\left(\frac{t}{2} \right)}}{\left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}}$$
2
2 cos (t)*tan(t)
sin (t) + --------------
cot(t)
$$\sin^{2}{\left(t \right)} + \frac{\cos^{2}{\left(t \right)} \tan{\left(t \right)}}{\cot{\left(t \right)}}$$
2 4
2 4*cos (t)*sin (t)
sin (t) + -----------------
2
sin (2*t)
$$\frac{4 \sin^{4}{\left(t \right)} \cos^{2}{\left(t \right)}}{\sin^{2}{\left(2 t \right)}} + \sin^{2}{\left(t \right)}$$
// 0 for t mod pi = 0\ || | 2*|< 2 | ||sin (t) otherwise | \\ /
$$2 \left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\sin^{2}{\left(t \right)} & \text{otherwise} \end{cases}\right)$$
2
/ 2/t pi\\ 2
|1 - cot |- + --|| *(1 + sin(t))
\ \2 4 //
---------------------------------
2
$$\frac{\left(- \cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\sin{\left(t \right)} + 1\right)^{2}}{2}$$
2 1 4*csc (2*t) ------- + --------------- 2 4 2 csc (t) csc (t)*sec (t)
$$\frac{1}{\csc^{2}{\left(t \right)}} + \frac{4 \csc^{2}{\left(2 t \right)}}{\csc^{4}{\left(t \right)} \sec^{2}{\left(t \right)}}$$
4 2/ pi\
4*sin (t)*sin |t + --|
2 \ 2 /
sin (t) + ----------------------
2
sin (2*t)
$$\frac{4 \sin^{4}{\left(t \right)} \sin^{2}{\left(t + \frac{\pi}{2} \right)}}{\sin^{2}{\left(2 t \right)}} + \sin^{2}{\left(t \right)}$$
2
/ 2/t pi\\
2*|-1 + tan |- + --||
\ \2 4 //
----------------------
2
/ 2/t pi\\
|1 + tan |- + --||
\ \2 4 //
$$\frac{2 \left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
2
/ 2/t pi\\
2*|1 - cot |- + --||
\ \2 4 //
---------------------
2
/ 2/t pi\\
|1 + cot |- + --||
\ \2 4 //
$$\frac{2 \left(- \cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}{\left(\cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
/1 cos(2*t)\
|- + --------|*tan(t)
1 cos(2*t) \2 2 /
- - -------- + ---------------------
2 2 cot(t)
$$\frac{\left(\frac{\cos{\left(2 t \right)}}{2} + \frac{1}{2}\right) \tan{\left(t \right)}}{\cot{\left(t \right)}} - \frac{\cos{\left(2 t \right)}}{2} + \frac{1}{2}$$
2
1 - cos(2*t) / 2/t\\ 4/t\ 2
------------ + |1 - tan |-|| *cos |-|*tan (t)
2 \ \2// \2/
$$\left(- \tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2} \cos^{4}{\left(\frac{t}{2} \right)} \tan^{2}{\left(t \right)} + \frac{- \cos{\left(2 t \right)} + 1}{2}$$
2 4/ pi\
4*cos (t)*cos |t - --|
2/ pi\ \ 2 /
cos |t - --| + ----------------------
\ 2 / 2/ pi\
cos |2*t - --|
\ 2 /
$$\frac{4 \cos^{2}{\left(t \right)} \cos^{4}{\left(t - \frac{\pi}{2} \right)}}{\cos^{2}{\left(2 t - \frac{\pi}{2} \right)}} + \cos^{2}{\left(t - \frac{\pi}{2} \right)}$$
2/ pi\
4*sec |2*t - --|
1 \ 2 /
------------ + --------------------
2/ pi\ 2 4/ pi\
sec |t - --| sec (t)*sec |t - --|
\ 2 / \ 2 /
$$\frac{1}{\sec^{2}{\left(t - \frac{\pi}{2} \right)}} + \frac{4 \sec^{2}{\left(2 t - \frac{\pi}{2} \right)}}{\sec^{2}{\left(t \right)} \sec^{4}{\left(t - \frac{\pi}{2} \right)}}$$
2
1 - cos(2*t) / 2/t\\ 4/t\ 2 2
------------ + |1 - tan |-|| *cos |-|*sec (t)*sin (t)
2 \ \2// \2/
$$\left(- \tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2} \sin^{2}{\left(t \right)} \cos^{4}{\left(\frac{t}{2} \right)} \sec^{2}{\left(t \right)} + \frac{- \cos{\left(2 t \right)} + 1}{2}$$
2 1 cos(2*t) / 2/t\\ 4/t\ 2 2 - - -------- + |1 - tan |-|| *cos |-|*sec (t)*sin (t) 2 2 \ \2// \2/
$$\left(- \tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2} \sin^{2}{\left(t \right)} \cos^{4}{\left(\frac{t}{2} \right)} \sec^{2}{\left(t \right)} - \frac{\cos{\left(2 t \right)}}{2} + \frac{1}{2}$$
// 0 for t mod pi = 0\ || | || 2/t\ | || 4*cot |-| | || \2/ | 2*|<-------------- otherwise | || 2 | ||/ 2/t\\ | |||1 + cot |-|| | ||\ \2// | \\ /
$$2 \left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{t}{2} \right)}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// / 3*pi\ \ || 1 for |t + ----| mod 2*pi = 0| || \ 2 / | 2*|< | || 4/t\ 2/t\ | ||- 4*cos |-| + 4*cos |-| otherwise | \\ \2/ \2/ /
$$2 \left(\begin{cases} 1 & \text{for}\: \left(t + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\- 4 \cos^{4}{\left(\frac{t}{2} \right)} + 4 \cos^{2}{\left(\frac{t}{2} \right)} & \text{otherwise} \end{cases}\right)$$
2 4 1 cos(2*t) / 2/t\\ / 2/t\\ 8/t\ 2 - - -------- + |1 - tan |-|| *|1 - tan |-|| *cos |-|*tan (t) 2 2 \ \2// \ \4// \4/
$$\left(- \tan^{2}{\left(\frac{t}{4} \right)} + 1\right)^{4} \left(- \tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2} \cos^{8}{\left(\frac{t}{4} \right)} \tan^{2}{\left(t \right)} - \frac{\cos{\left(2 t \right)}}{2} + \frac{1}{2}$$
/ 2 2 \
|1 cos (t) sin (t)|
2 2 |- + ------- - -------|*tan(t)
1 sin (t) cos (t) \2 2 2 /
- + ------- - ------- + ------------------------------
2 2 2 cot(t)
$$\frac{\sin^{2}{\left(t \right)}}{2} - \frac{\cos^{2}{\left(t \right)}}{2} + \frac{\left(- \frac{\sin^{2}{\left(t \right)}}{2} + \frac{\cos^{2}{\left(t \right)}}{2} + \frac{1}{2}\right) \tan{\left(t \right)}}{\cot{\left(t \right)}} + \frac{1}{2}$$
2
2/t\ / 2/t\\ 2
4*tan |-| |1 - tan |-|| *tan (t)
\2/ \ \2//
-------------- + ----------------------
2 2
/ 2/t\\ / 2/t\\
|1 + tan |-|| |1 + tan |-||
\ \2// \ \2//
$$\frac{\left(- \tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2} \tan^{2}{\left(t \right)}}{\left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} + \frac{4 \tan^{2}{\left(\frac{t}{2} \right)}}{\left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}}$$
2 2
/ /t\\ / /t\\ 4/t\
|1 + tan|-|| *|-1 + tan|-|| *cos |-|*(1 - cos(2*t))
1 - cos(2*t) \ \2// \ \2// \2/
------------ + ---------------------------------------------------
2 1 + cos(2*t)
$$\frac{\left(- \cos{\left(2 t \right)} + 1\right) \left(\tan{\left(\frac{t}{2} \right)} - 1\right)^{2} \left(\tan{\left(\frac{t}{2} \right)} + 1\right)^{2} \cos^{4}{\left(\frac{t}{2} \right)}}{\cos{\left(2 t \right)} + 1} + \frac{- \cos{\left(2 t \right)} + 1}{2}$$
2
/ 4/t\\
| 4*sin |-||
| \2/| 4 4/pi t\
/pi \ 4*|1 - ---------| *sin (t)*sin |-- + -|
sin|-- + 2*t| | 2 | \2 2/
1 \2 / \ sin (t) /
- - ------------- + ---------------------------------------
2 2 2
sin (2*t)
$$\frac{4 \left(- \frac{4 \sin^{4}{\left(\frac{t}{2} \right)}}{\sin^{2}{\left(t \right)}} + 1\right)^{2} \sin^{4}{\left(t \right)} \sin^{4}{\left(\frac{t}{2} + \frac{\pi}{2} \right)}}{\sin^{2}{\left(2 t \right)}} - \frac{\sin{\left(2 t + \frac{\pi}{2} \right)}}{2} + \frac{1}{2}$$
2
/ 2/t pi\\
| cos |- - --||
| \2 2 /| 4/t\ 2/ pi\
|1 - ------------| *cos |-|*cos |t - --|
| 2/t\ | \2/ \ 2 /
| cos |-| |
1 cos(2*t) \ \2/ /
- - -------- + ----------------------------------------
2 2 2
cos (t)
$$\frac{\left(1 - \frac{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \cos^{4}{\left(\frac{t}{2} \right)} \cos^{2}{\left(t - \frac{\pi}{2} \right)}}{\cos^{2}{\left(t \right)}} - \frac{\cos{\left(2 t \right)}}{2} + \frac{1}{2}$$
// 1 for t mod 2*pi = 0\ // 0 for t mod pi = 0\
2 || | || |
tan (t)*|< 2 | + |< 2 |
||cos (t) otherwise | ||sin (t) otherwise |
\\ / \\ /
$$\left(\left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\cos^{2}{\left(t \right)} & \text{otherwise} \end{cases}\right) \tan^{2}{\left(t \right)}\right) + \left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\sin^{2}{\left(t \right)} & \text{otherwise} \end{cases}\right)$$
// / 3*pi\ \ || 1 for |t + ----| mod 2*pi = 0| || \ 2 / | || | || 2 | ||/ 2/t pi\\ | 2*|<|-1 + tan |- + --|| | ||\ \2 4 // | ||-------------------- otherwise | || 2 | ||/ 2/t pi\\ | |||1 + tan |- + --|| | \\\ \2 4 // /
$$2 \left(\begin{cases} 1 & \text{for}\: \left(t + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)$$
2
/ 1 \
|1 - -------|
| 2/t\|
| cot |-||
4 \ \2//
---------------------- + ----------------------
2 2
/ 1 \ 2/t\ / 1 \ 2
|1 + -------| *cot |-| |1 + -------| *cot (t)
| 2/t\| \2/ | 2/t\|
| cot |-|| | cot |-||
\ \2// \ \2//
$$\frac{\left(1 - \frac{1}{\cot^{2}{\left(\frac{t}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \cot^{2}{\left(t \right)}} + \frac{4}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \cot^{2}{\left(\frac{t}{2} \right)}}$$
2
/ 2/t\ \
| sec |-| |
| \2/ | 2
|1 - ------------| *sec (t)
| 2/t pi\|
| sec |- - --||
1 1 \ \2 2 //
- - ---------- + ---------------------------
2 2*sec(2*t) 4/t\ 2/ pi\
sec |-|*sec |t - --|
\2/ \ 2 /
$$\frac{1}{2} - \frac{1}{2 \sec{\left(2 t \right)}} + \frac{\left(- \frac{\sec^{2}{\left(\frac{t}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec^{2}{\left(t \right)}}{\sec^{4}{\left(\frac{t}{2} \right)} \sec^{2}{\left(t - \frac{\pi}{2} \right)}}$$
2 4
/ 2/t\\ / 2/t\\ 2
2 |1 - tan |-|| *|1 - tan |-|| *tan (t)
1 1 - tan (t) \ \2// \ \4//
- - --------------- + -------------------------------------
2 / 2 \ 4
2*\1 + tan (t)/ / 2/t\\
|1 + tan |-||
\ \4//
$$\frac{\left(- \tan^{2}{\left(\frac{t}{4} \right)} + 1\right)^{4} \left(- \tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2} \tan^{2}{\left(t \right)}}{\left(\tan^{2}{\left(\frac{t}{4} \right)} + 1\right)^{4}} - \frac{- \tan^{2}{\left(t \right)} + 1}{2 \left(\tan^{2}{\left(t \right)} + 1\right)} + \frac{1}{2}$$
2
/ 2/pi t\\
| csc |-- - -||
| \2 2/| 2/pi \
|1 - ------------| *csc |-- - t|
| 2/t\ | \2 /
| csc |-| |
1 1 \ \2/ /
- - --------------- + --------------------------------
2 /pi \ 2 4/pi t\
2*csc|-- - 2*t| csc (t)*csc |-- - -|
\2 / \2 2/
$$\frac{1}{2} - \frac{1}{2 \csc{\left(- 2 t + \frac{\pi}{2} \right)}} + \frac{\left(1 - \frac{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \csc^{2}{\left(- t + \frac{\pi}{2} \right)}}{\csc^{2}{\left(t \right)} \csc^{4}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}}$$
// 1 for t mod 2*pi = 0\
4 || |
4*sin (t)*|< 2/ pi\ |
||sin |t + --| otherwise | // 0 for t mod pi = 0\
\\ \ 2 / / || |
--------------------------------------------- + |< 2 |
2 ||sin (t) otherwise |
sin (2*t) \\ /
$$\left(\frac{4 \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\sin^{2}{\left(t + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \sin^{4}{\left(t \right)}}{\sin^{2}{\left(2 t \right)}}\right) + \left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\sin^{2}{\left(t \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for t mod 2*pi = 0\
2/ pi\ || |
cos |t - --|*|< 2 |
\ 2 / ||cos (t) otherwise | // 0 for t mod pi = 0\
\\ / || |
------------------------------------------- + |< 2/ pi\ |
2 ||cos |t - --| otherwise |
cos (t) \\ \ 2 / /
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\cos^{2}{\left(t - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\frac{\left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\cos^{2}{\left(t \right)} & \text{otherwise} \end{cases}\right) \cos^{2}{\left(t - \frac{\pi}{2} \right)}}{\cos^{2}{\left(t \right)}}\right)$$
// 1 for t mod 2*pi = 0\
|| |
2 || 1 |
sec (t)*|<------- otherwise |
|| 2 | // 0 for t mod pi = 0\
||sec (t) | || |
\\ / || 1 |
-------------------------------------- + |<------------ otherwise |
2/ pi\ || 2/ pi\ |
sec |t - --| ||sec |t - --| |
\ 2 / \\ \ 2 / /
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{1}{\sec^{2}{\left(t - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\frac{\left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{1}{\sec^{2}{\left(t \right)}} & \text{otherwise} \end{cases}\right) \sec^{2}{\left(t \right)}}{\sec^{2}{\left(t - \frac{\pi}{2} \right)}}\right)$$
// 1 for t mod 2*pi = 0\
|| |
2/pi \ || 1 |
csc |-- - t|*|<------------ otherwise |
\2 / || 2/pi \ | // 0 for t mod pi = 0\
||csc |-- - t| | || |
\\ \2 / / || 1 |
------------------------------------------------ + |<------- otherwise |
2 || 2 |
csc (t) ||csc (t) |
\\ /
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{1}{\csc^{2}{\left(t \right)}} & \text{otherwise} \end{cases}\right) + \left(\frac{\left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{1}{\csc^{2}{\left(- t + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \csc^{2}{\left(- t + \frac{\pi}{2} \right)}}{\csc^{2}{\left(t \right)}}\right)$$
2
2/t\ / 2 \ 4/t\ 2/t pi\
4*tan |-| 64*\1 + tan (t)/ *tan |-|*tan |- + --|
\2/ \2/ \2 4 /
-------------- + ------------------------------------------
2 4 2
/ 2/t\\ / 2/t\\ / 2/t pi\\ 2
|1 + tan |-|| |1 + tan |-|| *|1 + tan |- + --|| *tan (t)
\ \2// \ \2// \ \2 4 //
$$\frac{4 \tan^{2}{\left(\frac{t}{2} \right)}}{\left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} + \frac{64 \left(\tan^{2}{\left(t \right)} + 1\right)^{2} \tan^{4}{\left(\frac{t}{2} \right)} \tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{4} \left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \tan^{2}{\left(t \right)}}$$
2
/ 4/t\\
| 4*sin |-||
| \2/| 4
2 4/t\ 4*|1 - ---------| *sin (t)
16*sin (t)*sin |-| | 2 |
\2/ \ sin (t) /
---------------------- + --------------------------
2 2
/ 2 4/t\\ / 4/t\\
|sin (t) + 4*sin |-|| | 4*sin |-||
\ \2// | \2/| 2
|1 + ---------| *sin (2*t)
| 2 |
\ sin (t) /
$$\frac{16 \sin^{4}{\left(\frac{t}{2} \right)} \sin^{2}{\left(t \right)}}{\left(4 \sin^{4}{\left(\frac{t}{2} \right)} + \sin^{2}{\left(t \right)}\right)^{2}} + \frac{4 \left(- \frac{4 \sin^{4}{\left(\frac{t}{2} \right)}}{\sin^{2}{\left(t \right)}} + 1\right)^{2} \sin^{4}{\left(t \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{t}{2} \right)}}{\sin^{2}{\left(t \right)}} + 1\right)^{2} \sin^{2}{\left(2 t \right)}}$$
2
/ 4/t\\
| 4*sin |-||
| \2/| 4
4/t\ 4*|1 - ---------| *sin (t)
16*sin |-| | 2 |
\2/ \ sin (t) /
------------------------ + --------------------------
2 2
/ 4/t\\ / 4/t\\
| 4*sin |-|| | 4*sin |-||
| \2/| 2 | \2/| 2
|1 + ---------| *sin (t) |1 + ---------| *sin (2*t)
| 2 | | 2 |
\ sin (t) / \ sin (t) /
$$\frac{4 \left(- \frac{4 \sin^{4}{\left(\frac{t}{2} \right)}}{\sin^{2}{\left(t \right)}} + 1\right)^{2} \sin^{4}{\left(t \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{t}{2} \right)}}{\sin^{2}{\left(t \right)}} + 1\right)^{2} \sin^{2}{\left(2 t \right)}} + \frac{16 \sin^{4}{\left(\frac{t}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{t}{2} \right)}}{\sin^{2}{\left(t \right)}} + 1\right)^{2} \sin^{2}{\left(t \right)}}$$
// t \
/ 1 for t mod pi = 0 || 1 for - mod 2*pi = 0|
< 2 || 2 |
1 \cos(2*t) otherwise / 2/t\\ 2 || |
- - --------------------------- + |1 - tan |-|| *tan (t)*|< 4 |
2 2 \ \2// ||/ 2/t\\ 8/t\ |
|||-1 + cot |-|| *sin |-| otherwise |
||\ \4// \4/ |
\\ /
$$\left(\left(- \tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2} \left(\begin{cases} 1 & \text{for}\: \frac{t}{2} \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{t}{4} \right)} - 1\right)^{4} \sin^{8}{\left(\frac{t}{4} \right)} & \text{otherwise} \end{cases}\right) \tan^{2}{\left(t \right)}\right) - \left(\frac{\begin{cases} 1 & \text{for}\: t \bmod \pi = 0 \\\cos{\left(2 t \right)} & \text{otherwise} \end{cases}}{2}\right) + \frac{1}{2}$$
// 1 for t mod 2*pi = 0\ // 0 for t mod pi = 0\
|| | || |
2 ||/ 1 for t mod 2*pi = 0 | ||/ 0 for t mod pi = 0 |
tan (t)*|<| | + |<| |
||< 2 otherwise | ||< 2 otherwise |
|||cos (t) otherwise | |||sin (t) otherwise |
\\\ / \\\ /
$$\left(\left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\cos^{2}{\left(t \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \tan^{2}{\left(t \right)}\right) + \left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\sin^{2}{\left(t \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
/ 1 for t mod 2*pi = 0
|
| 2
|/ 2/t\\
||-1 + cot |-||
<\ \2//
|--------------- otherwise // 0 for t mod pi = 0\
| 2 || |
| / 2/t\\ || 2/t\ |
| |1 + cot |-|| || 4*cot |-| |
\ \ \2// || \2/ |
------------------------------------ + |<-------------- otherwise |
2 || 2 |
cot (t) ||/ 2/t\\ |
|||1 + cot |-|| |
||\ \2// |
\\ /
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{t}{2} \right)}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) + \left(\frac{\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{t}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}}{\cot^{2}{\left(t \right)}}\right)$$
// 1 for t mod 2*pi = 0\ // 0 for t mod pi = 0\
|| | || |
|| 2 | || 2/t\ |
||/ 2/t\\ | || 4*tan |-| |
2 |||1 - tan |-|| | || \2/ |
tan (t)*|<\ \2// | + |<-------------- otherwise |
||-------------- otherwise | || 2 |
|| 2 | ||/ 2/t\\ |
||/ 2/t\\ | |||1 + tan |-|| |
|||1 + tan |-|| | ||\ \2// |
\\\ \2// / \\ /
$$\left(\left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(- \tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \tan^{2}{\left(t \right)}\right) + \left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{4 \tan^{2}{\left(\frac{t}{2} \right)}}{\left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// 1 for t mod 2*pi = 0\
|| |
|| 2 | // 0 for t mod pi = 0\
||/ 1 \ | || |
|||-1 + -------| | || 4 |
||| 2/t\| | ||---------------------- otherwise |
2 ||| tan |-|| | || 2 |
tan (t)*|<\ \2// | + |
$$\left(\left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{t}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{t}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right) \tan^{2}{\left(t \right)}\right) + \left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{4}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \tan^{2}{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
2
/ 2/t pi\\
| cos |- - --||
| \2 2 /| 2/ pi\
|1 - ------------| *cos |t - --|
2/t pi\ | 2/t\ | \ 2 /
4*cos |- - --| | cos |-| |
\2 2 / \ \2/ /
--------------------------- + --------------------------------
2 2
/ 2/t pi\\ / 2/t pi\\
| cos |- - --|| | cos |- - --||
| \2 2 /| 2/t\ | \2 2 /| 2
|1 + ------------| *cos |-| |1 + ------------| *cos (t)
| 2/t\ | \2/ | 2/t\ |
| cos |-| | | cos |-| |
\ \2/ / \ \2/ /
$$\frac{\left(1 - \frac{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \cos^{2}{\left(t - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \cos^{2}{\left(t \right)}} + \frac{4 \cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \cos^{2}{\left(\frac{t}{2} \right)}}$$
// t \
|| 1 for - mod 2*pi = 0|
|| 2 |
|| |
2 || 4 |
/ 1 \ ||/ 2/t\\ |
/ 1 for t mod pi = 0 |1 - -------| *|<|-1 + cot |-|| |
| | 2/t\| ||\ \4// |
| 2 | cot |-|| ||--------------- otherwise |
<-1 + cot (t) \ \2// || 4 |
|------------ otherwise || / 2/t\\ |
| 2 || |1 + cot |-|| |
1 \1 + cot (t) \\ \ \4// /
- - ------------------------------- + -----------------------------------------------------
2 2 2
cot (t)
$$\left(\frac{\left(1 - \frac{1}{\cot^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \left(\begin{cases} 1 & \text{for}\: \frac{t}{2} \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{t}{4} \right)} - 1\right)^{4}}{\left(\cot^{2}{\left(\frac{t}{4} \right)} + 1\right)^{4}} & \text{otherwise} \end{cases}\right)}{\cot^{2}{\left(t \right)}}\right) - \left(\frac{\begin{cases} 1 & \text{for}\: t \bmod \pi = 0 \\\frac{\cot^{2}{\left(t \right)} - 1}{\cot^{2}{\left(t \right)} + 1} & \text{otherwise} \end{cases}}{2}\right) + \frac{1}{2}$$
2
/ 2/t\ \
| sec |-| |
| \2/ | 2
|1 - ------------| *sec (t)
2/t\ | 2/t pi\|
4*sec |-| | sec |- - --||
\2/ \ \2 2 //
-------------------------------- + --------------------------------
2 2
/ 2/t\ \ / 2/t\ \
| sec |-| | | sec |-| |
| \2/ | 2/t pi\ | \2/ | 2/ pi\
|1 + ------------| *sec |- - --| |1 + ------------| *sec |t - --|
| 2/t pi\| \2 2 / | 2/t pi\| \ 2 /
| sec |- - --|| | sec |- - --||
\ \2 2 // \ \2 2 //
$$\frac{\left(- \frac{\sec^{2}{\left(\frac{t}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec^{2}{\left(t \right)}}{\left(\frac{\sec^{2}{\left(\frac{t}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec^{2}{\left(t - \frac{\pi}{2} \right)}} + \frac{4 \sec^{2}{\left(\frac{t}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{t}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}$$
2
/ 2/pi t\\
| csc |-- - -||
| \2 2/| 2/pi \
|1 - ------------| *csc |-- - t|
2/pi t\ | 2/t\ | \2 /
4*csc |-- - -| | csc |-| |
\2 2/ \ \2/ /
--------------------------- + --------------------------------
2 2
/ 2/pi t\\ / 2/pi t\\
| csc |-- - -|| | csc |-- - -||
| \2 2/| 2/t\ | \2 2/| 2
|1 + ------------| *csc |-| |1 + ------------| *csc (t)
| 2/t\ | \2/ | 2/t\ |
| csc |-| | | csc |-| |
\ \2/ / \ \2/ /
$$\frac{\left(1 - \frac{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \csc^{2}{\left(- t + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \csc^{2}{\left(t \right)}} + \frac{4 \csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \csc^{2}{\left(\frac{t}{2} \right)}}$$
// 0 for t mod pi = 0\
// 1 for t mod 2*pi = 0\ || |
|| | || 2 |
|| 2 | || sin (t) |
||/ 2 4/t\\ | ||------------------------ otherwise |
2 |||sin (t) - 4*sin |-|| | || 2 |
tan (t)*|<\ \2// | + |
$$\left(\left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(- 4 \sin^{4}{\left(\frac{t}{2} \right)} + \sin^{2}{\left(t \right)}\right)^{2}}{\left(4 \sin^{4}{\left(\frac{t}{2} \right)} + \sin^{2}{\left(t \right)}\right)^{2}} & \text{otherwise} \end{cases}\right) \tan^{2}{\left(t \right)}\right) + \left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{\sin^{2}{\left(t \right)}}{\left(1 + \frac{\sin^{2}{\left(t \right)}}{4 \sin^{4}{\left(\frac{t}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
/ 1 for t mod 2*pi = 0
|
|/ 1 for t mod 2*pi = 0
||
|| 2
||/ 2/t\\
<||-1 + cot |-||
|<\ \2// otherwise // 0 for t mod pi = 0\
||--------------- otherwise || |
|| 2 ||/ 0 for t mod pi = 0 |
|| / 2/t\\ ||| |
|| |1 + cot |-|| ||| 2/t\ |
\\ \ \2// ||| 4*cot |-| |
--------------------------------------------------------- + |<| \2/ |
2 ||<-------------- otherwise otherwise |
cot (t) ||| 2 |
|||/ 2/t\\ |
||||1 + cot |-|| |
|||\ \2// |
\\\ /
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{t}{2} \right)}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \left(\frac{\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{t}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}}{\cot^{2}{\left(t \right)}}\right)$$
// 1 for t mod 2*pi = 0\
|| |
|| 2 |
||/ 2 \ |
||| sin (t) | |
|||-1 + ---------| |
||| 4/t\| |
4 ||| 4*sin |-|| |
4*sin (t)*|<\ \2// |
||----------------- otherwise |
|| 2 |
|| / 2 \ | // 0 for t mod pi = 0\
|| | sin (t) | | || |
|| |1 + ---------| | || 2 |
|| | 4/t\| | || sin (t) |
|| | 4*sin |-|| | ||------------------------ otherwise |
\\ \ \2// / || 2 |
-------------------------------------------------- + |
$$\left(\frac{4 \left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sin^{2}{\left(t \right)}}{4 \sin^{4}{\left(\frac{t}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sin^{2}{\left(t \right)}}{4 \sin^{4}{\left(\frac{t}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right) \sin^{4}{\left(t \right)}}{\sin^{2}{\left(2 t \right)}}\right) + \left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{\sin^{2}{\left(t \right)}}{\left(1 + \frac{\sin^{2}{\left(t \right)}}{4 \sin^{4}{\left(\frac{t}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// / pi\ \
// 0 for t mod pi = 0\ || 0 for |t + --| mod pi = 0| // zoo for 2*t mod pi = 0\ // 0 for t mod pi = 0\
|| | || \ 2 / | || | || |
4*|< 4/t\ 8/t\ |*|< |*|< 2 | + |< 2 |
||16*cot |-|*sin |-| otherwise | || 2 2/t pi\ | ||------------ otherwise | ||sin (t) otherwise |
\\ \2/ \2/ / ||(1 + sin(t)) *cot |- + --| otherwise | \\1 - cos(4*t) / \\ /
\\ \2 4 / /
$$\left(4 \left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\16 \sin^{8}{\left(\frac{t}{2} \right)} \cot^{4}{\left(\frac{t}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(t + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(t \right)} + 1\right)^{2} \cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: 2 t \bmod \pi = 0 \\\frac{2}{- \cos{\left(4 t \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\sin^{2}{\left(t \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for t mod 2*pi = 0\
|| |
|| 2 |
||/ 2/t\ \ |
||| cos |-| | |
||| \2/ | |
|||-1 + ------------| |
||| 2/t pi\| |
2/ pi\ ||| cos |- - --|| |
cos |t - --|*|<\ \2 2 // |
\ 2 / ||-------------------- otherwise |
|| 2 |
||/ 2/t\ \ | // 0 for t mod pi = 0\
||| cos |-| | | || |
||| \2/ | | || 2/t\ |
|||1 + ------------| | || 4*cos |-| |
||| 2/t pi\| | || \2/ |
||| cos |- - --|| | ||-------------------------------- otherwise |
\\\ \2 2 // / || 2 |
-------------------------------------------------------- + |
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{4 \cos^{2}{\left(\frac{t}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{t}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\frac{\left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(\frac{\cos^{2}{\left(\frac{t}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\cos^{2}{\left(\frac{t}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \cos^{2}{\left(t - \frac{\pi}{2} \right)}}{\cos^{2}{\left(t \right)}}\right)$$
// 1 for t mod 2*pi = 0\
|| |
|| 2 |
||/ 2/t pi\\ |
||| sec |- - --|| |
||| \2 2 /| |
|||-1 + ------------| |
||| 2/t\ | |
2 ||| sec |-| | |
sec (t)*|<\ \2/ / |
||-------------------- otherwise |
|| 2 |
||/ 2/t pi\\ | // 0 for t mod pi = 0\
||| sec |- - --|| | || |
||| \2 2 /| | || 2/t pi\ |
|||1 + ------------| | || 4*sec |- - --| |
||| 2/t\ | | || \2 2 / |
||| sec |-| | | ||--------------------------- otherwise |
\\\ \2/ / / || 2 |
--------------------------------------------------- + |
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{4 \sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \sec^{2}{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\frac{\left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right) \sec^{2}{\left(t \right)}}{\sec^{2}{\left(t - \frac{\pi}{2} \right)}}\right)$$
// 1 for t mod 2*pi = 0\
|| |
|| 2 |
||/ 2/t\ \ |
||| csc |-| | |
||| \2/ | |
|||-1 + ------------| |
||| 2/pi t\| |
2/pi \ ||| csc |-- - -|| |
csc |-- - t|*|<\ \2 2// |
\2 / ||-------------------- otherwise |
|| 2 |
||/ 2/t\ \ | // 0 for t mod pi = 0\
||| csc |-| | | || |
||| \2/ | | || 2/t\ |
|||1 + ------------| | || 4*csc |-| |
||| 2/pi t\| | || \2/ |
||| csc |-- - -|| | ||-------------------------------- otherwise |
\\\ \2 2// / || 2 |
-------------------------------------------------------- + |
$$\left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{4 \csc^{2}{\left(\frac{t}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{t}{2} \right)}}{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2} \csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\frac{\left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{t}{2} \right)}}{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\csc^{2}{\left(\frac{t}{2} \right)}}{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \csc^{2}{\left(- t + \frac{\pi}{2} \right)}}{\csc^{2}{\left(t \right)}}\right)$$
// / pi\ \
// 0 for t mod pi = 0\ || 0 for |t + --| mod pi = 0| // 0 for t mod pi = 0\
|| | || \ 2 / | // zoo for 2*t mod pi = 0\ || |
|| 4/t\ | || | || | || 2/t\ |
|| 16*cot |-| | || 2/t pi\ | || 2 | || 4*cot |-| |
|| \2/ | || 4*cot |- + --| | ||/ 2 \ | || \2/ |
4*|<-------------- otherwise |*|< \2 4 / |*|<\1 + cot (t)/ | + |<-------------- otherwise |
|| 4 | ||------------------- otherwise | ||-------------- otherwise | || 2 |
||/ 2/t\\ | || 2 | || 2 | ||/ 2/t\\ |
|||1 + cot |-|| | ||/ 2/t pi\\ | || 4*cot (t) | |||1 + cot |-|| |
||\ \2// | |||1 + cot |- + --|| | \\ / ||\ \2// |
\\ / ||\ \2 4 // | \\ /
\\ /
$$\left(4 \left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{16 \cot^{4}{\left(\frac{t}{2} \right)}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{4}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(t + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)}}{\left(\cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: 2 t \bmod \pi = 0 \\\frac{\left(\cot^{2}{\left(t \right)} + 1\right)^{2}}{4 \cot^{2}{\left(t \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{t}{2} \right)}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)$$
4*Piecewise((0, Mod(t = pi, 0)), (16*cot(t/2)^4/(1 + cot(t/2)^2)^4, True))*Piecewise((0, Mod(t + pi/2 = pi, 0)), (4*cot(t/2 + pi/4)^2/(1 + cot(t/2 + pi/4)^2)^2, True))*Piecewise((±oo, Mod(2*t = pi, 0)), ((1 + cot(t)^2)^2/(4*cot(t)^2), True)) + Piecewise((0, Mod(t = pi, 0)), (4*cot(t/2)^2/(1 + cot(t/2)^2)^2, True))
Комбинаторика
[src]
2 2
cos (t)*tan(t) + sin (t)*cot(t)
-------------------------------
cot(t)
$$\frac{\sin^{2}{\left(t \right)} \cot{\left(t \right)} + \cos^{2}{\left(t \right)} \tan{\left(t \right)}}{\cot{\left(t \right)}}$$
(cos(t)^2*tan(t) + sin(t)^2*cot(t))/cot(t)
Рациональный знаменатель
[src]
2
2 cos (t)*tan(t)
sin (t) + --------------
cot(t)
$$\sin^{2}{\left(t \right)} + \frac{\cos^{2}{\left(t \right)} \tan{\left(t \right)}}{\cot{\left(t \right)}}$$
2 2
cos (t)*tan(t) + sin (t)*cot(t)
-------------------------------
cot(t)
$$\frac{\sin^{2}{\left(t \right)} \cot{\left(t \right)} + \cos^{2}{\left(t \right)} \tan{\left(t \right)}}{\cot{\left(t \right)}}$$
(cos(t)^2*tan(t) + sin(t)^2*cot(t))/cot(t)