Господин Экзамен
Другие калькуляторы
-
Как пользоваться?
- cos(x)^2-3*sin(x)*cos(x)+2*sin(x)^2 если x=-4
- cos(4*a) если a=-1
- 21*a-20*a если a=1/3
- 4*d-9+3*d если d=-1/3
- Разложить многочлен на множители:
- 18*x*y-6*x^2
- 144-y^2/12-y
- a^4+1024
- 64-125*y^3
- Общий знаменатель:
- a-c/c-a-c/a+c
- c2/(c3-8)-(5*c+1)/(8-c3)-(3-3*c)/(8-c3)
- 8/15*3/5+2/3*3/5+3/5-8/15*3/5
- 7^n+1+7^n/4^n+1*2^n/49^n
- Умножение столбиком:
- 96*18
- 654*645
- 806*128
- 11*21
- Деление столбиком:
- 48048/6
- 232/3
- 412/3
- 7410/3
- Раскрыть скобки в:
- (x-2)^4+4*(x-2)^3-16*(x-2)+11
- (d+4)^3-(d-1)*(d+3)
- 6*(x-7)
- (c+b)*(c-b)-(5*c^2-b)^2
- Разложение числа на простые множители:
- 7776
- 2475
- 1225
- 3548
- Производная:
- -4
- Интеграл d{x}:
-
-4
- График функции y =:
- -4
-
Идентичные выражения
- cos(x)^ два - три *sin(x)*cos(x)+ два *sin(x)^ два если x=- четыре
- косинус от (x) в квадрате минус 3 умножить на синус от (x) умножить на косинус от (x) плюс 2 умножить на синус от (x) в квадрате если x равно минус 4
- косинус от (x) в степени два минус три умножить на синус от (x) умножить на косинус от (x) плюс два умножить на синус от (x) в степени два если x равно минус четыре
- cos(x)2-3*sin(x)*cos(x)+2*sin(x)2 если x=-4
- cosx2-3*sinx*cosx+2*sinx2 если x=-4
- cos(x)²-3*sin(x)*cos(x)+2*sin(x)² если x=-4
- cos(x) в степени 2-3*sin(x)*cos(x)+2*sin(x) в степени 2 если x=-4
- cos(x)^2-3sin(x)cos(x)+2sin(x)^2 если x=-4
- cos(x)2-3sin(x)cos(x)+2sin(x)2 если x=-4
- cosx2-3sinxcosx+2sinx2 если x=-4
- cosx^2-3sinxcosx+2sinx^2 если x=-4
- cos(x)^2-3*sin(x)*cos(x)+2*sin(x)^2 при x=-4
-
Похожие выражения
- cos(x)^2-3*sin(x)*cos(x)+2*sin(x)^2 если x=+4
- cos(x)^2-3*sin(x)*cos(x)-2*sin(x)^2 если x=-4
- cos(x)^2+3*sin(x)*cos(x)+2*sin(x)^2 если x=-4
- cosx^2-3*sinx*cosx+2*sinx^2 если x=-4
cos(x)^2-3*sin(x)*cos(x)+2*sin(x)^2 если x=-4
Решение
Вы ввели
[src]
2 2 cos (x) - 3*sin(x)*cos(x) + 2*sin (x)
$$2 \sin^{2}{\left(x \right)} - 3 \sin{\left(x \right)} \cos{\left(x \right)} + \cos^{2}{\left(x \right)}$$
cos(x)^2 - 3*sin(x)*cos(x) + 2*sin(x)^2
Общее упрощение
[src]
3 3*sin(2*x) cos(2*x) - - ---------- - -------- 2 2 2
$$- \frac{3 \sin{\left(2 x \right)}}{2} - \frac{\cos{\left(2 x \right)}}{2} + \frac{3}{2}$$
3/2 - 3*sin(2*x)/2 - cos(2*x)/2
Подстановка условия
[src]
cos(x)^2 - 3*sin(x)*cos(x) + 2*sin(x)^2 при x = -4
подставляем
2 2 cos (x) - 3*sin(x)*cos(x) + 2*sin (x)
$$2 \sin^{2}{\left(x \right)} - 3 \sin{\left(x \right)} \cos{\left(x \right)} + \cos^{2}{\left(x \right)}$$
3 3*sin(2*x) cos(2*x) - - ---------- - -------- 2 2 2
$$- \frac{3 \sin{\left(2 x \right)}}{2} - \frac{\cos{\left(2 x \right)}}{2} + \frac{3}{2}$$
переменные
x = -4
$$x = -4$$
3 3*sin(2*(-4)) cos(2*(-4)) - - ------------- - ----------- 2 2 2
$$- \frac{3 \sin{\left(2 (-4) \right)}}{2} - \frac{\cos{\left(2 (-4) \right)}}{2} + \frac{3}{2}$$
3 cos(8) 3*sin(8) - - ------ + -------- 2 2 2
$$- \frac{\cos{\left(8 \right)}}{2} + \frac{3 \sin{\left(8 \right)}}{2} + \frac{3}{2}$$
3/2 - cos(8)/2 + 3*sin(8)/2
Собрать выражение
[src]
3 3*sin(2*x) cos(2*x) - - ---------- - -------- 2 2 2
$$- \frac{3 \sin{\left(2 x \right)}}{2} - \frac{\cos{\left(2 x \right)}}{2} + \frac{3}{2}$$
3/2 - 3*sin(2*x)/2 - cos(2*x)/2
Комбинаторика
[src]
(-sin(x) + cos(x))*(-2*sin(x) + cos(x))
$$\left(- 2 \sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right)$$
(-sin(x) + cos(x))*(-2*sin(x) + cos(x))
Степени
[src]
/ I*x -I*x\
2 2 |e e | / -I*x I*x\
/ I*x -I*x\ / -I*x I*x\ 3*I*|---- + -----|*\- e + e /
|e e | \- e + e / \ 2 2 /
|---- + -----| - ----------------- + -----------------------------------
\ 2 2 / 2 2
$$\left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{2} + \frac{3 i \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right) \left(e^{i x} - e^{- i x}\right)}{2} - \frac{\left(e^{i x} - e^{- i x}\right)^{2}}{2}$$
(exp(i*x)/2 + exp(-i*x)/2)^2 - (-exp(-i*x) + exp(i*x))^2/2 + 3*i*(exp(i*x)/2 + exp(-i*x)/2)*(-exp(-i*x) + exp(i*x))/2
Тригонометрическая часть
[src]
3 3*sin(2*x) cos(2*x) - - ---------- - -------- 2 2 2
$$- \frac{3 \sin{\left(2 x \right)}}{2} - \frac{\cos{\left(2 x \right)}}{2} + \frac{3}{2}$$
3 cos(2*x) - - -------- - 3*cos(x)*sin(x) 2 2
$$- 3 \sin{\left(x \right)} \cos{\left(x \right)} - \frac{\cos{\left(2 x \right)}}{2} + \frac{3}{2}$$
/ pi\
3*cos|2*x - --|
3 \ 2 / cos(2*x)
- - --------------- - --------
2 2 2
$$- \frac{\cos{\left(2 x \right)}}{2} - \frac{3 \cos{\left(2 x - \frac{\pi}{2} \right)}}{2} + \frac{3}{2}$$
/pi \
sin|-- + 2*x|
3 3*sin(2*x) \2 /
- - ---------- - -------------
2 2 2
$$- \frac{3 \sin{\left(2 x \right)}}{2} - \frac{\sin{\left(2 x + \frac{\pi}{2} \right)}}{2} + \frac{3}{2}$$
3 3 1
- - --------------- - ----------
2 / pi\ 2*sec(2*x)
2*sec|2*x - --|
\ 2 /
$$\frac{3}{2} - \frac{3}{2 \sec{\left(2 x - \frac{\pi}{2} \right)}} - \frac{1}{2 \sec{\left(2 x \right)}}$$
3 3 1
- - ---------- - ---------------
2 2*csc(2*x) /pi \
2*csc|-- - 2*x|
\2 /
$$\frac{3}{2} - \frac{1}{2 \csc{\left(- 2 x + \frac{\pi}{2} \right)}} - \frac{3}{2 \csc{\left(2 x \right)}}$$
1 2 3 ------- + ------- - ------------- 2 2 csc(x)*sec(x) sec (x) csc (x)
$$\frac{1}{\sec^{2}{\left(x \right)}} - \frac{3}{\csc{\left(x \right)} \sec{\left(x \right)}} + \frac{2}{\csc^{2}{\left(x \right)}}$$
2/ pi\ 2 / pi\
sin |x + --| + 2*sin (x) - 3*sin(x)*sin|x + --|
\ 2 / \ 2 /
$$2 \sin^{2}{\left(x \right)} - 3 \sin{\left(x \right)} \sin{\left(x + \frac{\pi}{2} \right)} + \sin^{2}{\left(x + \frac{\pi}{2} \right)}$$
2 2 3 sin (x) cos (x) - + ------- - ------- - 3*cos(x)*sin(x) 2 2 2
$$\frac{\sin^{2}{\left(x \right)}}{2} - 3 \sin{\left(x \right)} \cos{\left(x \right)} - \frac{\cos^{2}{\left(x \right)}}{2} + \frac{3}{2}$$
2 2/ pi\ / pi\
cos (x) + 2*cos |x - --| - 3*cos(x)*cos|x - --|
\ 2 / \ 2 /
$$\cos^{2}{\left(x \right)} - 3 \cos{\left(x \right)} \cos{\left(x - \frac{\pi}{2} \right)} + 2 \cos^{2}{\left(x - \frac{\pi}{2} \right)}$$
2 /x\
1 + cos (x) - cos(2*x) - 3*(1 + cos(x))*cos(x)*tan|-|
\2/
$$- 3 \left(\cos{\left(x \right)} + 1\right) \cos{\left(x \right)} \tan{\left(\frac{x}{2} \right)} + \cos^{2}{\left(x \right)} - \cos{\left(2 x \right)} + 1$$
1 2 3
------- + ------------ - ------------------
2 2/ pi\ / pi\
sec (x) sec |x - --| sec(x)*sec|x - --|
\ 2 / \ 2 /
$$\frac{2}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} - \frac{3}{\sec{\left(x \right)} \sec{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{\sec^{2}{\left(x \right)}}$$
9 cos(2*x) 2 3*sin(2*x) - + -------- - 4*cos(x) - 2*(1 - cos(x)) - ---------- 2 2 2
$$- 2 \left(- \cos{\left(x \right)} + 1\right)^{2} - \frac{3 \sin{\left(2 x \right)}}{2} - 4 \cos{\left(x \right)} + \frac{\cos{\left(2 x \right)}}{2} + \frac{9}{2}$$
2
3 3*tan(x) 1 - tan (x)
- - ----------- - ---------------
2 2 / 2 \
1 + tan (x) 2*\1 + tan (x)/
$$- \frac{- \tan^{2}{\left(x \right)} + 1}{2 \left(\tan^{2}{\left(x \right)} + 1\right)} + \frac{3}{2} - \frac{3 \tan{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1}$$
1 2 3
------- + ------------ - ------------------
2 2/pi \ /pi \
sec (x) sec |-- - x| sec(x)*sec|-- - x|
\2 / \2 /
$$\frac{2}{\sec^{2}{\left(- x + \frac{\pi}{2} \right)}} - \frac{3}{\sec{\left(x \right)} \sec{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{\sec^{2}{\left(x \right)}}$$
1 2 3
------------ + ------- - ------------------
2/pi \ 2 /pi \
csc |-- - x| csc (x) csc(x)*csc|-- - x|
\2 / \2 /
$$\frac{1}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} - \frac{3}{\csc{\left(x \right)} \csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{2}{\csc^{2}{\left(x \right)}}$$
1 2 3
------------ + ------------ - -----------------------
2/pi \ 2 /pi \
csc |-- - x| csc (pi - x) csc(pi - x)*csc|-- - x|
\2 / \2 /
$$\frac{1}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} - \frac{3}{\csc{\left(- x + \pi \right)} \csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{2}{\csc^{2}{\left(- x + \pi \right)}}$$
2/x pi\
4*tan |- + --|
/ 2/x\\ \2 4 /
1 - cos(2*x) - 3*|-1 + 2*cos |-||*sin(x) + -------------------
\ \2// 2
/ 2/x pi\\
|1 + tan |- + --||
\ \2 4 //
$$- 3 \cdot \left(2 \cos^{2}{\left(\frac{x}{2} \right)} - 1\right) \sin{\left(x \right)} - \cos{\left(2 x \right)} + 1 + \frac{4 \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
// 0 for 2*x mod pi = 0\ / 1 for x mod pi = 0
3*|< | <
3 \\sin(2*x) otherwise / \cos(2*x) otherwise
- - --------------------------------- - ---------------------------
2 2 2
$$\left(- \frac{3 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases}\right)}{2}\right) - \left(\frac{\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\cos{\left(2 x \right)} & \text{otherwise} \end{cases}}{2}\right) + \frac{3}{2}$$
4 8/x\ 3 2/x\ 2 4/x\ 6/x\
sin (x) + 16*sin |-| - 12*sin (x)*sin |-| + 24*sin (x)*sin |-| + 48*sin |-|*sin(x)
\2/ \2/ \2/ \2/
----------------------------------------------------------------------------------
2
/ 2 4/x\\
|sin (x) + 4*sin |-||
\ \2//
$$\frac{16 \sin^{8}{\left(\frac{x}{2} \right)} + 48 \sin^{6}{\left(\frac{x}{2} \right)} \sin{\left(x \right)} + 24 \sin^{4}{\left(\frac{x}{2} \right)} \sin^{2}{\left(x \right)} - 12 \sin^{2}{\left(\frac{x}{2} \right)} \sin^{3}{\left(x \right)} + \sin^{4}{\left(x \right)}}{\left(4 \sin^{4}{\left(\frac{x}{2} \right)} + \sin^{2}{\left(x \right)}\right)^{2}}$$
// 0 for 2*x mod pi = 0\ / 1 for x mod pi = 0
|| | |
|| 2*cot(x) | | 2
3*|<----------- otherwise | <-1 + cot (x)
|| 2 | |------------ otherwise
||1 + cot (x) | | 2
3 \\ / \1 + cot (x)
- - ------------------------------------ - -------------------------------
2 2 2
$$\left(- \frac{3 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}\right)}{2}\right) - \left(\frac{\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}}{2}\right) + \frac{3}{2}$$
2
/ 2/x pi\\ 2 2/x\ / 2/x pi\\ / 2/x\\
|1 - cot |- + --|| *(1 + sin(x)) 3*cos |-|*|1 - cot |- + --||*|1 - tan |-||*(1 + sin(x))
1 + cos(2*x) \ \2 4 // \2/ \ \2 4 // \ \2//
------------ + --------------------------------- - -------------------------------------------------------
2 2 2
$$- \frac{3 \cdot \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(x \right)} + 1\right) \cos^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{\left(- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\sin{\left(x \right)} + 1\right)^{2}}{2} + \frac{\cos{\left(2 x \right)} + 1}{2}$$
2
/ 2/x\\ 2/x\ / 2/x\\ /x\
|1 - tan |-|| 8*tan |-| 6*|1 - tan |-||*tan|-|
\ \2// \2/ \ \2// \2/
-------------- + -------------- - ----------------------
2 2 2
/ 2/x\\ / 2/x\\ / 2/x\\
|1 + tan |-|| |1 + tan |-|| |1 + tan |-||
\ \2// \ \2// \ \2//
$$\frac{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} - \frac{6 \cdot \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \tan{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + \frac{8 \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$
2
/ 1 \ / 1 \
|1 - -------| 6*|1 - -------|
| 2/x\| | 2/x\|
| cot |-|| | cot |-||
\ \2// 8 \ \2//
-------------- + ---------------------- - ---------------------
2 2 2
/ 1 \ / 1 \ 2/x\ / 1 \ /x\
|1 + -------| |1 + -------| *cot |-| |1 + -------| *cot|-|
| 2/x\| | 2/x\| \2/ | 2/x\| \2/
| cot |-|| | cot |-|| | cot |-||
\ \2// \ \2// \ \2//
$$\frac{\left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{2}} - \frac{6 \cdot \left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \cot{\left(\frac{x}{2} \right)}} + \frac{8}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \cot^{2}{\left(\frac{x}{2} \right)}}$$
2/x pi\ 2/x\ /x\ /x pi\
4*tan |- + --| 8*tan |-| 12*tan|-|*tan|- + --|
\2 4 / \2/ \2/ \2 4 /
------------------- + -------------- - --------------------------------
2 2 / 2/x\\ / 2/x pi\\
/ 2/x pi\\ / 2/x\\ |1 + tan |-||*|1 + tan |- + --||
|1 + tan |- + --|| |1 + tan |-|| \ \2// \ \2 4 //
\ \2 4 // \ \2//
$$\frac{4 \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} - \frac{12 \tan{\left(\frac{x}{2} \right)} \tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)} + \frac{8 \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$
2/x pi\ 2/x\ /x\ /x pi\
4*tan |- + --| 8*cot |-| 12*cot|-|*tan|- + --|
\2 4 / \2/ \2/ \2 4 /
------------------- + -------------- - --------------------------------
2 2 / 2/x\\ / 2/x pi\\
/ 2/x pi\\ / 2/x\\ |1 + cot |-||*|1 + tan |- + --||
|1 + tan |- + --|| |1 + cot |-|| \ \2// \ \2 4 //
\ \2 4 // \ \2//
$$\frac{8 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} - \frac{12 \tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)} \cot{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \frac{4 \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ || | // 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ || | 2*|< 2 | - 3*|< |*|< | + |< 2 | ||sin (x) otherwise | \\sin(x) otherwise / \\cos(x) otherwise / ||cos (x) otherwise | \\ / \\ /
$$\left(- 3 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)$$
2 2
/ 2/x\\ / 2/x pi\\ / 2/x\\ / 2/x pi\\
|-1 + cot |-|| 2*|-1 + tan |- + --|| 3*|-1 + cot |-||*|-1 + tan |- + --||
\ \2// \ \2 4 // \ \2// \ \2 4 //
--------------- + ---------------------- - ------------------------------------
2 2 / 2/x\\ / 2/x pi\\
/ 2/x\\ / 2/x pi\\ |1 + cot |-||*|1 + tan |- + --||
|1 + cot |-|| |1 + tan |- + --|| \ \2// \ \2 4 //
\ \2// \ \2 4 //
$$\frac{2 \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} - \frac{3 \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$
// 0 for x mod pi = 0\ // 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ || | || | // 1 for x mod 2*pi = 0\ || | 2*|< 2/ pi\ | - 3*|< / pi\ |*|< | + |< 2 | ||cos |x - --| otherwise | ||cos|x - --| otherwise | \\cos(x) otherwise / ||cos (x) otherwise | \\ \ 2 / / \\ \ 2 / / \\ /
$$\left(- 3 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\cos{\left(x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\cos^{2}{\left(x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ // 1 for x mod 2*pi = 0\ || | // 0 for x mod pi = 0\ || | || | 2*|< 2 | - 3*|< |*|< / pi\ | + |< 2/ pi\ | ||sin (x) otherwise | \\sin(x) otherwise / ||sin|x + --| otherwise | ||sin |x + --| otherwise | \\ / \\ \ 2 / / \\ \ 2 / /
$$\left(- 3 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\sin{\left(x + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\sin^{2}{\left(x + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)$$
2 2
/ 2/x\\ / 2/x pi\\ / 2/x pi\\ / 2/x\\
|1 - tan |-|| 2*|1 - cot |- + --|| 3*|1 - cot |- + --||*|1 - tan |-||
\ \2// \ \2 4 // \ \2 4 // \ \2//
-------------- + --------------------- - ----------------------------------
2 2 / 2/x pi\\ / 2/x\\
/ 2/x\\ / 2/x pi\\ |1 + cot |- + --||*|1 + tan |-||
|1 + tan |-|| |1 + cot |- + --|| \ \2 4 // \ \2//
\ \2// \ \2 4 //
$$\frac{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} - \frac{3 \cdot \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)} + \frac{2 \left(- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
// 0 for x mod pi = 0\
// 0 for x mod pi = 0\ || | // 1 for x mod 2*pi = 0\
|| | ||1 - cos(x) | // 1 for x mod 2*pi = 0\ || |
2*|< 2 | - 3*|<---------- otherwise |*|< | + |< 2 |
||sin (x) otherwise | || /x\ | \\cos(x) otherwise / ||cos (x) otherwise |
\\ / || tan|-| | \\ /
\\ \2/ /
$$\left(- 3 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{- \cos{\left(x \right)} + 1}{\tan{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)$$
// 0 for x mod pi = 0\ // 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ || | || | // 1 for x mod 2*pi = 0\ || | || 1 | || 1 | || | || 1 | 2*|<------------ otherwise | - 3*|<----------- otherwise |*|< 1 | + |<------- otherwise | || 2/ pi\ | || / pi\ | ||------ otherwise | || 2 | ||sec |x - --| | ||sec|x - --| | \\sec(x) / ||sec (x) | \\ \ 2 / / \\ \ 2 / / \\ /
$$\left(- 3 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(x \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{1}{\sec^{2}{\left(x \right)}} & \text{otherwise} \end{cases}\right)$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ // 1 for x mod 2*pi = 0\ || | // 0 for x mod pi = 0\ || | || | || 1 | || | || 1 | || 1 | 2*|<------- otherwise | - 3*|< 1 |*|<----------- otherwise | + |<------------ otherwise | || 2 | ||------ otherwise | || /pi \ | || 2/pi \ | ||csc (x) | \\csc(x) / ||csc|-- - x| | ||csc |-- - x| | \\ / \\ \2 / / \\ \2 / /
$$\left(- 3 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\csc{\left(x \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\csc^{2}{\left(x \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{1}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
2
/ 4/x\\ / 4/x\\
| 4*sin |-|| | 4*sin |-||
| \2/| 2/x\ | \2/|
|1 - ---------| 4/x\ 12*sin |-|*|1 - ---------|
| 2 | 32*sin |-| \2/ | 2 |
\ sin (x) / \2/ \ sin (x) /
---------------- + ------------------------ - --------------------------
2 2 2
/ 4/x\\ / 4/x\\ / 4/x\\
| 4*sin |-|| | 4*sin |-|| | 4*sin |-||
| \2/| | \2/| 2 | \2/|
|1 + ---------| |1 + ---------| *sin (x) |1 + ---------| *sin(x)
| 2 | | 2 | | 2 |
\ sin (x) / \ sin (x) / \ sin (x) /
$$\frac{\left(- \frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{2}}{\left(\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{2}} - \frac{12 \left(- \frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right) \sin^{2}{\left(\frac{x}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{2} \sin{\left(x \right)}} + \frac{32 \sin^{4}{\left(\frac{x}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{2} \sin^{2}{\left(x \right)}}$$
// / 3*pi\ \ || 1 for |x + ----| mod 2*pi = 0| // / 3*pi\ \ // 1 for x mod 2*pi = 0\ || \ 2 / | // 1 for x mod 2*pi = 0\ || 1 for |x + ----| mod 2*pi = 0| || | 2*|< | - 3*|< |*|< \ 2 / | + |< 2 | || 4/x\ 2/x\ | \\cos(x) otherwise / || | ||cos (x) otherwise | ||- 4*cos |-| + 4*cos |-| otherwise | \\sin(x) otherwise / \\ / \\ \2/ \2/ /
$$\left(- 3 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\- 4 \cos^{4}{\left(\frac{x}{2} \right)} + 4 \cos^{2}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right)\right)$$
// / pi\ \ // / pi\ \
// 0 for x mod pi = 0\ || 0 for |x + --| mod pi = 0| || 0 for |x + --| mod pi = 0|
|| | // 0 for x mod pi = 0\ || \ 2 / | || \ 2 / |
2*|< 2 | - 3*|< |*|< | + |< |
||sin (x) otherwise | \\sin(x) otherwise / || /x pi\ | || 2 2/x pi\ |
\\ / ||(1 + sin(x))*cot|- + --| otherwise | ||(1 + sin(x)) *cot |- + --| otherwise |
\\ \2 4 / / \\ \2 4 / /
$$\left(- 3 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(x \right)} + 1\right) \cot{\left(\frac{x}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \left(x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(x \right)} + 1\right)^{2} \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)$$
2
/ 2/x\ \ / 2/x\ \
| sec |-| | | sec |-| |
| \2/ | | \2/ | /x\
|1 - ------------| 6*|1 - ------------|*sec|-|
| 2/x pi\| 2/x\ | 2/x pi\| \2/
| sec |- - --|| 8*sec |-| | sec |- - --||
\ \2 2 // \2/ \ \2 2 //
------------------- + -------------------------------- - -------------------------------
2 2 2
/ 2/x\ \ / 2/x\ \ / 2/x\ \
| sec |-| | | sec |-| | | sec |-| |
| \2/ | | \2/ | 2/x pi\ | \2/ | /x pi\
|1 + ------------| |1 + ------------| *sec |- - --| |1 + ------------| *sec|- - --|
| 2/x pi\| | 2/x pi\| \2 2 / | 2/x pi\| \2 2 /
| sec |- - --|| | sec |- - --|| | sec |- - --||
\ \2 2 // \ \2 2 // \ \2 2 //
$$\frac{\left(- \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}}{\left(\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}} - \frac{6 \left(- \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{x}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + \frac{8 \sec^{2}{\left(\frac{x}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}$$
2
/ 2/x pi\\ / 2/x pi\\
| cos |- - --|| | cos |- - --||
| \2 2 /| | \2 2 /| /x pi\
|1 - ------------| 6*|1 - ------------|*cos|- - --|
| 2/x\ | 2/x pi\ | 2/x\ | \2 2 /
| cos |-| | 8*cos |- - --| | cos |-| |
\ \2/ / \2 2 / \ \2/ /
------------------- + --------------------------- - --------------------------------
2 2 2
/ 2/x pi\\ / 2/x pi\\ / 2/x pi\\
| cos |- - --|| | cos |- - --|| | cos |- - --||
| \2 2 /| | \2 2 /| 2/x\ | \2 2 /| /x\
|1 + ------------| |1 + ------------| *cos |-| |1 + ------------| *cos|-|
| 2/x\ | | 2/x\ | \2/ | 2/x\ | \2/
| cos |-| | | cos |-| | | cos |-| |
\ \2/ / \ \2/ / \ \2/ /
$$\frac{\left(1 - \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{2}}{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{2}} - \frac{6 \cdot \left(1 - \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right) \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \cos{\left(\frac{x}{2} \right)}} + \frac{8 \cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \cos^{2}{\left(\frac{x}{2} \right)}}$$
2
/ 2/pi x\\ / 2/pi x\\
| csc |-- - -|| | csc |-- - -||
| \2 2/| | \2 2/| /pi x\
|1 - ------------| 6*|1 - ------------|*csc|-- - -|
| 2/x\ | 2/pi x\ | 2/x\ | \2 2/
| csc |-| | 8*csc |-- - -| | csc |-| |
\ \2/ / \2 2/ \ \2/ /
------------------- + --------------------------- - --------------------------------
2 2 2
/ 2/pi x\\ / 2/pi x\\ / 2/pi x\\
| csc |-- - -|| | csc |-- - -|| | csc |-- - -||
| \2 2/| | \2 2/| 2/x\ | \2 2/| /x\
|1 + ------------| |1 + ------------| *csc |-| |1 + ------------| *csc|-|
| 2/x\ | | 2/x\ | \2/ | 2/x\ | \2/
| csc |-| | | csc |-| | | csc |-| |
\ \2/ / \ \2/ / \ \2/ /
$$\frac{\left(1 - \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{2}}{\left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{2}} - \frac{6 \cdot \left(1 - \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right) \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \csc{\left(\frac{x}{2} \right)}} + \frac{8 \csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \csc^{2}{\left(\frac{x}{2} \right)}}$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ || | // 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ || | || 2/x\ | || | || | || 2 | || 4*cot |-| | || /x\ | || 2/x\ | ||/ 2/x\\ | || \2/ | || 2*cot|-| | ||-1 + cot |-| | |||-1 + cot |-|| | 2*|<-------------- otherwise | - 3*|< \2/ |*|< \2/ | + |<\ \2// | || 2 | ||----------- otherwise | ||------------ otherwise | ||--------------- otherwise | ||/ 2/x\\ | || 2/x\ | || 2/x\ | || 2 | |||1 + cot |-|| | ||1 + cot |-| | ||1 + cot |-| | || / 2/x\\ | ||\ \2// | \\ \2/ / \\ \2/ / || |1 + cot |-|| | \\ / \\ \ \2// /
$$\left(- 3 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ || | // 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ || | || 2/x\ | || | || | || 2 | || 4*tan |-| | || /x\ | || 2/x\ | ||/ 2/x\\ | || \2/ | || 2*tan|-| | ||1 - tan |-| | |||1 - tan |-|| | 2*|<-------------- otherwise | - 3*|< \2/ |*|< \2/ | + |<\ \2// | || 2 | ||----------- otherwise | ||----------- otherwise | ||-------------- otherwise | ||/ 2/x\\ | || 2/x\ | || 2/x\ | || 2 | |||1 + tan |-|| | ||1 + tan |-| | ||1 + tan |-| | ||/ 2/x\\ | ||\ \2// | \\ \2/ / \\ \2/ / |||1 + tan |-|| | \\ / \\\ \2// /
$$\left(- 3 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{x}{2} \right)} + 1}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\
|| | // 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ || |
||/ 0 for x mod pi = 0 | || | || | ||/ 1 for x mod 2*pi = 0 |
2*|<| | - 3*|
$$\left(- 3 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\
// 1 for x mod 2*pi = 0\ || |
// 0 for x mod pi = 0\ || | || 2 |
|| | // 0 for x mod pi = 0\ || 1 | ||/ 1 \ |
|| 4 | || | ||-1 + ------- | |||-1 + -------| |
||---------------------- otherwise | || 2 | || 2/x\ | ||| 2/x\| |
|| 2 | ||-------------------- otherwise | || tan |-| | ||| tan |-|| |
2*|
$$\left(- 3 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right) \tan{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \tan^{2}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// / pi\ \
// 0 for x mod pi = 0\ // / pi\ \ || 0 for |x + --| mod pi = 0|
|| | // 0 for x mod pi = 0\ || 0 for |x + --| mod pi = 0| || \ 2 / |
|| 2/x\ | || | || \ 2 / | || |
|| 4*cot |-| | || /x\ | || | || 2/x pi\ |
|| \2/ | || 2*cot|-| | || /x pi\ | || 4*cot |- + --| |
2*|<-------------- otherwise | - 3*|< \2/ |*|< 2*cot|- + --| | + |< \2 4 / |
|| 2 | ||----------- otherwise | || \2 4 / | ||------------------- otherwise |
||/ 2/x\\ | || 2/x\ | ||---------------- otherwise | || 2 |
|||1 + cot |-|| | ||1 + cot |-| | || 2/x pi\ | ||/ 2/x pi\\ |
||\ \2// | \\ \2/ / ||1 + cot |- + --| | |||1 + cot |- + --|| |
\\ / \\ \2 4 / / ||\ \2 4 // |
\\ /
$$\left(- 3 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \left(x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// / 3*pi\ \ || 1 for |x + ----| mod 2*pi = 0| // / 3*pi\ \ // 1 for x mod 2*pi = 0\ || \ 2 / | // 1 for x mod 2*pi = 0\ || 1 for |x + ----| mod 2*pi = 0| || | || | || | || \ 2 / | || 2 | || 2 | || 2/x\ | || | ||/ 2/x\\ | ||/ 2/x pi\\ | ||-1 + cot |-| | || 2/x pi\ | |||-1 + cot |-|| | 2*|<|-1 + tan |- + --|| | - 3*|< \2/ |*|<-1 + tan |- + --| | + |<\ \2// | ||\ \2 4 // | ||------------ otherwise | || \2 4 / | ||--------------- otherwise | ||-------------------- otherwise | || 2/x\ | ||----------------- otherwise | || 2 | || 2 | ||1 + cot |-| | || 2/x pi\ | || / 2/x\\ | ||/ 2/x pi\\ | \\ \2/ / || 1 + tan |- + --| | || |1 + cot |-|| | |||1 + tan |- + --|| | \\ \2 4 / / \\ \ \2// / \\\ \2 4 // /
$$\left(- 3 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for x mod pi = 0\
|| | // 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\
|| 2 | || | || |
|| sin (x) | || 2*sin(x) | // 1 for x mod 2*pi = 0\ || 2 |
||------------------------ otherwise | ||---------------------------- otherwise | || | ||/ 2 4/x\\ |
|| 2 | || / 2 \ | || 2 | |||sin (x) - 4*sin |-|| |
2*|
$$\left(- 3 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \sin{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right) \left(- \cos{\left(x \right)} + 1\right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{4 \sin^{2}{\left(x \right)} + 4 \cos{\left(x \right)} - 4}{2 \left(- \cos{\left(x \right)} + 1\right)^{2} + 2 \sin^{2}{\left(x \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{\sin^{2}{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(- 4 \sin^{4}{\left(\frac{x}{2} \right)} + \sin^{2}{\left(x \right)}\right)^{2}}{\left(4 \sin^{4}{\left(\frac{x}{2} \right)} + \sin^{2}{\left(x \right)}\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\
// 1 for x mod 2*pi = 0\ || |
// 0 for x mod pi = 0\ || | || 2 |
|| | // 0 for x mod pi = 0\ || 2 | ||/ 2 \ |
|| 2 | || | || sin (x) | ||| sin (x) | |
|| sin (x) | || sin(x) | ||-1 + --------- | |||-1 + ---------| |
||------------------------ otherwise | ||----------------------- otherwise | || 4/x\ | ||| 4/x\| |
|| 2 | ||/ 2 \ | || 4*sin |-| | ||| 4*sin |-|| |
2*|
$$\left(- 3 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{\sin{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right) \sin^{2}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}}{1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{\sin^{2}{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ || | // 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ || | ||/ 0 for x mod pi = 0 | || | || | ||/ 1 for x mod 2*pi = 0 | ||| | ||/ 0 for x mod pi = 0 | ||/ 1 for x mod 2*pi = 0 | ||| | ||| 2/x\ | ||| | ||| | ||| 2 | ||| 4*cot |-| | ||| /x\ | ||| 2/x\ | |||/ 2/x\\ | 2*|<| \2/ | - 3*|<| 2*cot|-| |*|<|-1 + cot |-| | + |<||-1 + cot |-|| | ||<-------------- otherwise otherwise | ||< \2/ otherwise | ||< \2/ otherwise | ||<\ \2// otherwise | ||| 2 | |||----------- otherwise | |||------------ otherwise | |||--------------- otherwise | |||/ 2/x\\ | ||| 2/x\ | ||| 2/x\ | ||| 2 | ||||1 + cot |-|| | |||1 + cot |-| | |||1 + cot |-| | ||| / 2/x\\ | |||\ \2// | \\\ \2/ / \\\ \2/ / ||| |1 + cot |-|| | \\\ / \\\ \ \2// /
$$\left(- 3 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\
// 1 for x mod 2*pi = 0\ || |
// 0 for x mod pi = 0\ || | || 2 |
|| | // 0 for x mod pi = 0\ || 2/x\ | ||/ 2/x\ \ |
|| 2/x\ | || | || cos |-| | ||| cos |-| | |
|| 4*cos |-| | || /x\ | || \2/ | ||| \2/ | |
|| \2/ | || 2*cos|-| | ||-1 + ------------ | |||-1 + ------------| |
||-------------------------------- otherwise | || \2/ | || 2/x pi\ | ||| 2/x pi\| |
|| 2 | ||------------------------------ otherwise | || cos |- - --| | ||| cos |- - --|| |
2*|
$$\left(- 3 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{x}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \cos^{2}{\left(\frac{x}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\
// 1 for x mod 2*pi = 0\ || |
// 0 for x mod pi = 0\ || | || 2 |
|| | // 0 for x mod pi = 0\ || 2/x pi\ | ||/ 2/x pi\\ |
|| 2/x pi\ | || | || sec |- - --| | ||| sec |- - --|| |
|| 4*sec |- - --| | || /x pi\ | || \2 2 / | ||| \2 2 /| |
|| \2 2 / | || 2*sec|- - --| | ||-1 + ------------ | |||-1 + ------------| |
||--------------------------- otherwise | || \2 2 / | || 2/x\ | ||| 2/x\ | |
|| 2 | ||------------------------- otherwise | || sec |-| | ||| sec |-| | |
2*|
$$\left(- 3 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right) \sec{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \sec^{2}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\
// 1 for x mod 2*pi = 0\ || |
// 0 for x mod pi = 0\ || | || 2 |
|| | // 0 for x mod pi = 0\ || 2/x\ | ||/ 2/x\ \ |
|| 2/x\ | || | || csc |-| | ||| csc |-| | |
|| 4*csc |-| | || /x\ | || \2/ | ||| \2/ | |
|| \2/ | || 2*csc|-| | ||-1 + ------------ | |||-1 + ------------| |
||-------------------------------- otherwise | || \2/ | || 2/pi x\ | ||| 2/pi x\| |
|| 2 | ||------------------------------ otherwise | || csc |-- - -| | ||| csc |-- - -|| |
2*|
$$\left(- 3 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{x}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \csc^{2}{\left(\frac{x}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2} \csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)$$
2*Piecewise((0, Mod(x = pi, 0)), (4*csc(x/2)^2/((1 + csc(x/2)^2/csc(pi/2 - x/2)^2)^2*csc(pi/2 - x/2)^2), True)) - 3*Piecewise((0, Mod(x = pi, 0)), (2*csc(x/2)/((1 + csc(x/2)^2/csc(pi/2 - x/2)^2)*csc(pi/2 - x/2)), True))*Piecewise((1, Mod(x = 2*pi, 0)), ((-1 + csc(x/2)^2/csc(pi/2 - x/2)^2)/(1 + csc(x/2)^2/csc(pi/2 - x/2)^2), True)) + Piecewise((1, Mod(x = 2*pi, 0)), ((-1 + csc(x/2)^2/csc(pi/2 - x/2)^2)^2/(1 + csc(x/2)^2/csc(pi/2 - x/2)^2)^2, True))
Численный ответ
[src]
cos(x)^2 + 2.0*sin(x)^2 - 3.0*cos(x)*sin(x)
cos(x)^2 + 2.0*sin(x)^2 - 3.0*cos(x)*sin(x)