Господин Экзамен
Другие калькуляторы
-
Как пользоваться?
- (sqrt(x))^2 если x=1
- sin(2*x)/cos(x)+2 если x=-4
- sin(45+x)-cos(45+x) если x=-3
- k^3-12*k^2+48*k-64 если k=-3
- Разложить многочлен на множители:
- -x^2-2*x+1
- 16*x^3+8*x^2+x
- 36*x^2+6*x+9
- 121*p^2+25-110*p
- Общий знаменатель:
- b^2-4*b*y/2*y^2-b*y-4*y/b-2*y
- (3-2*m)/(m-7*n)+(4-2*m)/(m^2-49*n^2)
- y^2-1/y^2+5*y+6*y+2/y+1-y-1/y+3
- 2*sin(2*a)+cos(3*n/2-a)-sin(n+a)/1+sin(3*n/2-a)
- Умножение столбиком:
- 54*1000
- 1420*99
- 2886*46
- 613*148
- Деление столбиком:
- 36750/2
- 38418/2
- 38400/2
- 39016/2
- Раскрыть скобки в:
- (y-1)*(y^2-6*y+4)+y^3+7*y^2+8
- (2-5*d^2)*(25*d^4+10*d^2+4)
- 14*y-8*(8*y-9)
- (1+4*x-4*x^2)^135*(1+2*x)^5*(1-3*x+x^2+2*x^3)^145
- Разложение числа на простые множители:
- 43562
- 37842
- 54416
- 19241
- Производная:
- -1/3
- Интеграл d{x}:
-
-1/3
-
Идентичные выражения
- cos(x)+cos(два *x)- один если x=- один / три
- косинус от (x) плюс косинус от (2 умножить на x) минус 1 если x равно минус 1 делить на 3
- косинус от (x) плюс косинус от (два умножить на x) минус один если x равно минус один делить на три
- cos(x)+cos(2x)-1 если x=-1/3
- cosx+cos2x-1 если x=-1/3
- cos(x)+cos(2*x)-1 при x=-1/3
- cos(x)+cos(2*x)-1 если x=-1 разделить на 3
-
Похожие выражения
- cos(x)-cos(2*x)-1 если x=-1/3
- cos(x)+cos(2*x)+1 если x=-1/3
- cos(x)+cos(2*x)-1 если x=+1/3
- cosx+cos(2*x)-1 если x=-1/3
cos(x)+cos(2*x)-1 если x=-1/3
Решение
Вы ввели
[src]
cos(x) + cos(2*x) - 1
$$\cos{\left(x \right)} + \cos{\left(2 x \right)} - 1$$
cos(x) + cos(2*x) - 1*1
Подстановка условия
[src]
cos(x) + cos(2*x) - 1*1 при x = -1/3
подставляем
cos(x) + cos(2*x) - 1
$$\cos{\left(x \right)} + \cos{\left(2 x \right)} - 1$$
-1 + cos(x) + cos(2*x)
$$\cos{\left(x \right)} + \cos{\left(2 x \right)} - 1$$
переменные
x = -1/3
$$x = - \frac{1}{3}$$
-1 + cos((-1/3)) + cos(2*(-1/3))
$$\cos{\left((-1/3) \right)} + \cos{\left(2 (-1/3) \right)} - 1$$
-1 + cos(-1/3) + cos(2*-1/3)
$$-1 + \cos{\left(2 \left(- \frac{1}{3}\right) \right)} + \cos{\left(- \frac{1}{3} \right)}$$
-1 + cos(1/3) + cos(2/3)
$$-1 + \cos{\left(\frac{2}{3} \right)} + \cos{\left(\frac{1}{3} \right)}$$
-1 + cos(1/3) + cos(2/3)
Степени
[src]
I*x -I*x -2*I*x 2*I*x
e e e e
-1 + ---- + ----- + ------- + ------
2 2 2 2
$$\frac{e^{2 i x}}{2} + \frac{e^{i x}}{2} - 1 + \frac{e^{- i x}}{2} + \frac{e^{- 2 i x}}{2}$$
-1 + exp(i*x)/2 + exp(-i*x)/2 + exp(-2*i*x)/2 + exp(2*i*x)/2
Тригонометрическая часть
[src]
/x\ /3*x\
-1 + 2*cos|-|*cos|---|
\2/ \ 2 /
$$2 \cos{\left(\frac{x}{2} \right)} \cos{\left(\frac{3 x}{2} \right)} - 1$$
2 2/x\
-3 + 2*cos (x) + 2*cos |-|
\2/
$$2 \cos^{2}{\left(\frac{x}{2} \right)} + 2 \cos^{2}{\left(x \right)} - 3$$
1 1
-1 + ------ + --------
sec(x) sec(2*x)
$$-1 + \frac{1}{\sec{\left(2 x \right)}} + \frac{1}{\sec{\left(x \right)}}$$
/ pi\ /pi \
-1 + sin|x + --| + sin|-- + 2*x|
\ 2 / \2 /
$$\sin{\left(x + \frac{\pi}{2} \right)} + \sin{\left(2 x + \frac{\pi}{2} \right)} - 1$$
2 2 -1 + cos (x) - sin (x) + cos(x)
$$- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} + \cos{\left(x \right)} - 1$$
1 1
-1 + ----------- + -------------
/pi \ /pi \
csc|-- - x| csc|-- - 2*x|
\2 / \2 /
$$-1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}}$$
3 - cos(2*x) - 3*cos(x) + cos(3*x)
----------------------------------
2*(-1 + cos(x))
$$\frac{- 3 \cos{\left(x \right)} - \cos{\left(2 x \right)} + \cos{\left(3 x \right)} + 3}{2 \left(\cos{\left(x \right)} - 1\right)}$$
2/x\
2 -1 + cot |-|
-1 + cot (x) \2/
-1 + ------------ + ------------
2 2/x\
1 + cot (x) 1 + cot |-|
\2/
$$\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} - 1$$
2/x\
2 1 - tan |-|
1 - tan (x) \2/
-1 + ----------- + -----------
2 2/x\
1 + tan (x) 1 + tan |-|
\2/
$$\frac{- \tan^{2}{\left(\frac{x}{2} \right)} + 1}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{- \tan^{2}{\left(x \right)} + 1}{\tan^{2}{\left(x \right)} + 1} - 1$$
1
1 1 - -------
1 - ------- 2/x\
2 cot |-|
cot (x) \2/
-1 + ----------- + -----------
1 1
1 + ------- 1 + -------
2 2/x\
cot (x) cot |-|
\2/
$$\frac{1 - \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}}{1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}} + \frac{1 - \frac{1}{\cot^{2}{\left(x \right)}}}{1 + \frac{1}{\cot^{2}{\left(x \right)}}} - 1$$
/ pi\ /x pi\
2*tan|x + --| 2*tan|- + --|
\ 4 / \2 4 /
-1 + ---------------- + ----------------
2/ pi\ 2/x pi\
1 + tan |x + --| 1 + tan |- + --|
\ 4 / \2 4 /
$$-1 + \frac{2 \tan{\left(x + \frac{\pi}{4} \right)}}{\tan^{2}{\left(x + \frac{\pi}{4} \right)} + 1} + \frac{2 \tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1}$$
// 1 for x mod pi = 0\ // 1 for x mod 2*pi = 0\
-1 + |< | + |< |
\\cos(2*x) otherwise / \\cos(x) otherwise /
$$\left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\cos{\left(2 x \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases}\right) - 1$$
// 1 for x mod pi = 0\ // 1 for x mod 2*pi = 0\
|| | || |
-1 + |< 1 | + |< 1 |
||-------- otherwise | ||------ otherwise |
\\sec(2*x) / \\sec(x) /
$$\left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\sec{\left(2 x \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(x \right)}} & \text{otherwise} \end{cases}\right) - 1$$
// 1 for x mod pi = 0\ // 1 for x mod 2*pi = 0\
|| | || |
-1 + |< /pi \ | + |< / pi\ |
||sin|-- + 2*x| otherwise | ||sin|x + --| otherwise |
\\ \2 / / \\ \ 2 / /
$$\left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(2 x + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\sin{\left(x + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) - 1$$
// 1 for x mod pi = 0\ // 1 for x mod 2*pi = 0\
|| | || |
|| 1 | || 1 |
-1 + |<------------- otherwise | + |<----------- otherwise |
|| /pi \ | || /pi \ |
||csc|-- - 2*x| | ||csc|-- - x| |
\\ \2 / / \\ \2 / /
$$\left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) - 1$$
4/x\
4*sin |-| 4
\2/ 4*sin (x)
1 - --------- 1 - ---------
2 2
sin (x) sin (2*x)
-1 + ------------- + -------------
4/x\ 4
4*sin |-| 4*sin (x)
\2/ 1 + ---------
1 + --------- 2
2 sin (2*x)
sin (x)
$$\frac{- \frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1}{\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1} + \frac{- \frac{4 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} + 1}{\frac{4 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} + 1} - 1$$
// 1 for x mod 2*pi = 0\
// 1 for x mod pi = 0\ || |
|| | || 2/x\ |
|| 2 | ||-1 + cot |-| |
-1 + |<-1 + cot (x) | + |< \2/ |
||------------ otherwise | ||------------ otherwise |
|| 2 | || 2/x\ |
\\1 + cot (x) / ||1 + cot |-| |
\\ \2/ /
$$\left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) - 1$$
// 1 for x mod 2*pi = 0\
// 1 for x mod pi = 0\ || |
|| | || 2/x\ |
|| 2 | ||1 - tan |-| |
-1 + |<1 - tan (x) | + |< \2/ |
||----------- otherwise | ||----------- otherwise |
|| 2 | || 2/x\ |
\\1 + tan (x) / ||1 + tan |-| |
\\ \2/ /
$$\left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{- \tan^{2}{\left(x \right)} + 1}{\tan^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{x}{2} \right)} + 1}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) - 1$$
// 1 for x mod 2*pi = 0\
|| |
// 1 for x mod pi = 0\ || 2 |
-1 + |< | + |< -4 + 4*sin (x) + 4*cos(x) |
\\cos(2*x) otherwise / ||--------------------------- otherwise |
|| 2 2 |
\\2*(1 - cos(x)) + 2*sin (x) /
$$\left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\cos{\left(2 x \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{4 \sin^{2}{\left(x \right)} + 4 \cos{\left(x \right)} - 4}{2 \left(- \cos{\left(x \right)} + 1\right)^{2} + 2 \sin^{2}{\left(x \right)}} & \text{otherwise} \end{cases}\right) - 1$$
// / pi\ \
|| 0 for |x + --| mod pi = 0| // /pi \ \
|| \ 2 / | || 0 for |-- + 2*x| mod pi = 0|
-1 + |< | + |< \2 / |
|| /x pi\ | || |
||(1 + sin(x))*cot|- + --| otherwise | \\cos(2*x) otherwise /
\\ \2 4 / /
$$\left(\begin{cases} 0 & \text{for}\: \left(x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(x \right)} + 1\right) \cot{\left(\frac{x}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(2 x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\cos{\left(2 x \right)} & \text{otherwise} \end{cases}\right) - 1$$
2/x pi\
2/ pi\ cos |- - --|
cos |x - --| \2 2 /
\ 2 / 1 - ------------
1 - ------------ 2/x\
2 cos |-|
cos (x) \2/
-1 + ---------------- + ----------------
2/ pi\ 2/x pi\
cos |x - --| cos |- - --|
\ 2 / \2 2 /
1 + ------------ 1 + ------------
2 2/x\
cos (x) cos |-|
\2/
$$\frac{1 - \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}}{1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}} + \frac{1 - \frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}}}{1 + \frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}}} - 1$$
2/x\
2 sec |-|
sec (x) \2/
1 - ------------ 1 - ------------
2/ pi\ 2/x pi\
sec |x - --| sec |- - --|
\ 2 / \2 2 /
-1 + ---------------- + ----------------
2 2/x\
sec (x) sec |-|
1 + ------------ \2/
2/ pi\ 1 + ------------
sec |x - --| 2/x pi\
\ 2 / sec |- - --|
\2 2 /
$$\frac{- \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1}{\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1} + \frac{- \frac{\sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} + 1}{\frac{\sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} + 1} - 1$$
2/pi x\
2/pi \ csc |-- - -|
csc |-- - x| \2 2/
\2 / 1 - ------------
1 - ------------ 2/x\
2 csc |-|
csc (x) \2/
-1 + ---------------- + ----------------
2/pi \ 2/pi x\
csc |-- - x| csc |-- - -|
\2 / \2 2/
1 + ------------ 1 + ------------
2 2/x\
csc (x) csc |-|
\2/
$$\frac{1 - \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}}{1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}} + \frac{1 - \frac{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}}}{1 + \frac{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}}} - 1$$
// 1 for x mod 2*pi = 0\
// 1 for x mod pi = 0\ || |
|| | || 1 |
|| 1 | ||-1 + ------- |
||-1 + ------- | || 2/x\ |
|| 2 | || tan |-| |
-1 + |< tan (x) | + |< \2/ |
||------------ otherwise | ||------------ otherwise |
|| 1 | || 1 |
||1 + ------- | ||1 + ------- |
|| 2 | || 2/x\ |
\\ tan (x) / || tan |-| |
\\ \2/ /
$$\left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(x \right)}}}{1 + \frac{1}{\tan^{2}{\left(x \right)}}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}} & \text{otherwise} \end{cases}\right) - 1$$
// 1 for x mod pi = 0\ // 1 for x mod 2*pi = 0\
|| | || |
-1 + |
$$\left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\cos{\left(2 x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) - 1$$
// / pi\ \ // /pi \ \
|| 0 for |x + --| mod pi = 0| || 0 for |-- + 2*x| mod pi = 0|
|| \ 2 / | || \2 / |
|| | || |
|| /x pi\ | || / pi\ |
-1 + |< 2*cot|- + --| | + |< 2*cot|x + --| |
|| \2 4 / | || \ 4 / |
||---------------- otherwise | ||---------------- otherwise |
|| 2/x pi\ | || 2/ pi\ |
||1 + cot |- + --| | ||1 + cot |x + --| |
\\ \2 4 / / \\ \ 4 / /
$$\left(\begin{cases} 0 & \text{for}\: \left(x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(2 x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(x + \frac{\pi}{4} \right)}}{\cot^{2}{\left(x + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) - 1$$
// 1 for x mod 2*pi = 0\
// 1 for x mod pi = 0\ || |
|| | || 2 |
|| 2 | || sin (x) |
|| sin (2*x) | ||-1 + --------- |
||-1 + --------- | || 4/x\ |
|| 4 | || 4*sin |-| |
-1 + |< 4*sin (x) | + |< \2/ |
||-------------- otherwise | ||-------------- otherwise |
|| 2 | || 2 |
|| sin (2*x) | || sin (x) |
||1 + --------- | ||1 + --------- |
|| 4 | || 4/x\ |
\\ 4*sin (x) / || 4*sin |-| |
\\ \2/ /
$$\left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(2 x \right)}}{4 \sin^{4}{\left(x \right)}}}{1 + \frac{\sin^{2}{\left(2 x \right)}}{4 \sin^{4}{\left(x \right)}}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}}{1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}} & \text{otherwise} \end{cases}\right) - 1$$
// 1 for x mod 2*pi = 0\
// 1 for x mod pi = 0\ || |
|| | ||/ 1 for x mod 2*pi = 0 |
||/ 1 for x mod pi = 0 | ||| |
||| | ||| 2/x\ |
-1 + |<| 2 | + |<|-1 + cot |-| |
||<-1 + cot (x) otherwise | ||< \2/ otherwise |
|||------------ otherwise | |||------------ otherwise |
||| 2 | ||| 2/x\ |
\\\1 + cot (x) / |||1 + cot |-| |
\\\ \2/ /
$$\left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) - 1$$
// 1 for x mod 2*pi = 0\
// 1 for x mod pi = 0\ || |
|| | || 2/x\ |
|| 2 | || cos |-| |
|| cos (x) | || \2/ |
||-1 + ------------ | ||-1 + ------------ |
|| 2/ pi\ | || 2/x pi\ |
|| cos |x - --| | || cos |- - --| |
-1 + |< \ 2 / | + |< \2 2 / |
||----------------- otherwise | ||----------------- otherwise |
|| 2 | || 2/x\ |
|| cos (x) | || cos |-| |
|| 1 + ------------ | || \2/ |
|| 2/ pi\ | || 1 + ------------ |
|| cos |x - --| | || 2/x pi\ |
\\ \ 2 / / || cos |- - --| |
\\ \2 2 / /
$$\left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{\frac{\cos^{2}{\left(x \right)}}{\cos^{2}{\left(x - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(x \right)}}{\cos^{2}{\left(x - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) - 1$$
// 1 for x mod 2*pi = 0\
// 1 for x mod pi = 0\ || |
|| | || 2/x pi\ |
|| 2/ pi\ | || sec |- - --| |
|| sec |x - --| | || \2 2 / |
|| \ 2 / | ||-1 + ------------ |
||-1 + ------------ | || 2/x\ |
|| 2 | || sec |-| |
-1 + |< sec (x) | + |< \2/ |
||----------------- otherwise | ||----------------- otherwise |
|| 2/ pi\ | || 2/x pi\ |
|| sec |x - --| | || sec |- - --| |
|| \ 2 / | || \2 2 / |
|| 1 + ------------ | || 1 + ------------ |
|| 2 | || 2/x\ |
\\ sec (x) / || sec |-| |
\\ \2/ /
$$\left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(x - \frac{\pi}{2} \right)}}{\sec^{2}{\left(x \right)}}}{1 + \frac{\sec^{2}{\left(x - \frac{\pi}{2} \right)}}{\sec^{2}{\left(x \right)}}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}} & \text{otherwise} \end{cases}\right) - 1$$
// 1 for x mod 2*pi = 0\
// 1 for x mod pi = 0\ || |
|| | || 2/x\ |
|| 2 | || csc |-| |
|| csc (x) | || \2/ |
||-1 + ------------ | ||-1 + ------------ |
|| 2/pi \ | || 2/pi x\ |
|| csc |-- - x| | || csc |-- - -| |
-1 + |< \2 / | + |< \2 2/ |
||----------------- otherwise | ||----------------- otherwise |
|| 2 | || 2/x\ |
|| csc (x) | || csc |-| |
|| 1 + ------------ | || \2/ |
|| 2/pi \ | || 1 + ------------ |
|| csc |-- - x| | || 2/pi x\ |
\\ \2 / / || csc |-- - -| |
\\ \2 2/ /
$$\left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{\frac{\csc^{2}{\left(x \right)}}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(x \right)}}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) - 1$$
-1 + Piecewise((1, Mod(x = pi, 0)), ((-1 + csc(x)^2/csc(pi/2 - x)^2)/(1 + csc(x)^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))