Господин Экзамен
Другие калькуляторы
-
Как пользоваться?
- cos(2*a)-cos(a)^2 если a=-1/2
- (a+5)^2+(a+8)*(8-a) если a=-3
- x^2-6*x*y+10*y^2-4*y+7 если y=-2
- 36*a^2-24*a*b+4*b^2 если a=1
- Разложить многочлен на множители:
- 81*q^2-100*p^2
- s^2-r^2+10*s+25
- a^3+27
- 4*x^2-q^2
- Общий знаменатель:
- 1/b-1/a
- u+y/19*z-6*u+y/19*z
- cos(pi/7+a)*cos(5*pi/14-a)-sin(pi/7+a)*sin(5*pi/14)
- x-3/4*x^2+24*x+36/(x/3*x-9-3/x^2+3*x+x^2+9/27-3*x^2)
- Умножение столбиком:
- 38*42
- 16*38
- 22*45
- 815*204
- Деление столбиком:
- 122/6
- 183/9
- 183/4
- 648/2
- Раскрыть скобки в:
- 5*c*(c+3)*(c-3)
- (x+y+z)*(x-y-z)
- (7+3*y)*(3*y-7)
- (5*b+6)*(5*b-6)
- Разложение числа на простые множители:
- 300
- 20736
- 1260
- 7688
- График функции y =:
- -1/2
- Интеграл d{x}:
-
-1/2
- Производная:
- -1/2
-
Идентичные выражения
- cos(два *a)-cos(a)^ два если a=- один / два
- косинус от (2 умножить на a) минус косинус от (a) в квадрате если a равно минус 1 делить на 2
- косинус от (два умножить на a) минус косинус от (a) в степени два если a равно минус один делить на два
- cos(2*a)-cos(a)2 если a=-1/2
- cos2*a-cosa2 если a=-1/2
- cos(2*a)-cos(a)² если a=-1/2
- cos(2*a)-cos(a) в степени 2 если a=-1/2
- cos(2a)-cos(a)^2 если a=-1/2
- cos(2a)-cos(a)2 если a=-1/2
- cos2a-cosa2 если a=-1/2
- cos2a-cosa^2 если a=-1/2
- cos(2*a)-cos(a)^2 при a=-1/2
- cos(2*a)-cos(a)^2 если a=-1 разделить на 2
-
Похожие выражения
- cos(2*a)-cos(a)^2 если a=+1/2
- cos(2*a)+cos(a)^2 если a=-1/2
cos(2*a)-cos(a)^2 если a=-1/2
Решение
Вы ввели
[src]
2 cos(2*a) - cos (a)
$$- \cos^{2}{\left(a \right)} + \cos{\left(2 a \right)}$$
cos(2*a) - cos(a)^2
Подстановка условия
[src]
cos(2*a) - cos(a)^2 при a = -1/2
подставляем
2 cos(2*a) - cos (a)
$$- \cos^{2}{\left(a \right)} + \cos{\left(2 a \right)}$$
2 -sin (a)
$$- \sin^{2}{\left(a \right)}$$
переменные
a = -1/2
$$a = - \frac{1}{2}$$
2 -sin ((-1/2))
$$- \sin^{2}{\left((-1/2) \right)}$$
2 -sin (-1/2)
$$- \sin^{2}{\left(- \frac{1}{2} \right)}$$
2 -sin (1/2)
$$- \sin^{2}{\left(\frac{1}{2} \right)}$$
-sin(1/2)^2
Степени
[src]
2 -2*I*a 2*I*a / I*a -I*a\ e e |e e | ------- + ------ - |---- + -----| 2 2 \ 2 2 /
$$- \left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2}\right)^{2} + \frac{e^{2 i a}}{2} + \frac{e^{- 2 i a}}{2}$$
exp(-2*i*a)/2 + exp(2*i*a)/2 - (exp(i*a)/2 + exp(-i*a)/2)^2
Собрать выражение
[src]
1 cos(2*a) - - + -------- 2 2
$$\frac{\cos{\left(2 a \right)}}{2} - \frac{1}{2}$$
-1/2 + cos(2*a)/2
Тригонометрическая часть
[src]
2 -sin (a)
$$- \sin^{2}{\left(a \right)}$$
-1 ------- 2 csc (a)
$$- \frac{1}{\csc^{2}{\left(a \right)}}$$
2/ pi\
-cos |a - --|
\ 2 /
$$- \cos^{2}{\left(a - \frac{\pi}{2} \right)}$$
1 cos(2*a) - - + -------- 2 2
$$\frac{\cos{\left(2 a \right)}}{2} - \frac{1}{2}$$
-1
------------
2/ pi\
sec |a - --|
\ 2 /
$$- \frac{1}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}$$
1 1
-------- - -------
sec(2*a) 2
sec (a)
$$\frac{1}{\sec{\left(2 a \right)}} - \frac{1}{\sec^{2}{\left(a \right)}}$$
2/ pi\ /pi \
- sin |a + --| + sin|-- + 2*a|
\ 2 / \2 /
$$- \sin^{2}{\left(a + \frac{\pi}{2} \right)} + \sin{\left(2 a + \frac{\pi}{2} \right)}$$
2 2 1 cos (a) sin (a) - - + ------- - ------- 2 2 2
$$- \frac{\sin^{2}{\left(a \right)}}{2} + \frac{\cos^{2}{\left(a \right)}}{2} - \frac{1}{2}$$
2/a\
-4*tan |-|
\2/
--------------
2
/ 2/a\\
|1 + tan |-||
\ \2//
$$- \frac{4 \tan^{2}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}$$
// 0 for a mod pi = 0\ || | -|< 2 | ||sin (a) otherwise | \\ /
$$- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin^{2}{\left(a \right)} & \text{otherwise} \end{cases}$$
1 1 ------------- - ------------ /pi \ 2/pi \ csc|-- - 2*a| csc |-- - a| \2 / \2 /
$$\frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}} - \frac{1}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}}$$
// 0 for a mod pi = 0\ || | || 2/a\ | || 4*cot |-| | || \2/ | -|<-------------- otherwise | || 2 | ||/ 2/a\\ | |||1 + cot |-|| | ||\ \2// | \\ /
$$- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}$$
2
/ 2/a\\
2 |-1 + cot |-||
-1 + cot (a) \ \2//
------------ - ---------------
2 2
1 + cot (a) / 2/a\\
|1 + cot |-||
\ \2//
$$- \frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} + \frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1}$$
2
/ 2/a\\
2 |1 - tan |-||
1 - tan (a) \ \2//
----------- - --------------
2 2
1 + tan (a) / 2/a\\
|1 + tan |-||
\ \2//
$$- \frac{\left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} + \frac{- \tan^{2}{\left(a \right)} + 1}{\tan^{2}{\left(a \right)} + 1}$$
2
/ 4/a\\
| 4*sin |-||
| \2/|
|1 - ---------|
| 2 |
\ sin (a) /
- ---------------- + cos(2*a)
2
/ 4/a\\
| 4*sin |-||
| \2/|
|1 + ---------|
| 2 |
\ sin (a) /
$$\cos{\left(2 a \right)} - \frac{\left(- \frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right)^{2}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right)^{2}}$$
2/a pi\ / pi\
4*tan |- + --| 2*tan|a + --|
\2 4 / \ 4 /
- ------------------- + ----------------
2 2/ pi\
/ 2/a pi\\ 1 + tan |a + --|
|1 + tan |- + --|| \ 4 /
\ \2 4 //
$$\frac{2 \tan{\left(a + \frac{\pi}{4} \right)}}{\tan^{2}{\left(a + \frac{\pi}{4} \right)} + 1} - \frac{4 \tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
2
/ 1 \
1 |1 - -------|
1 - ------- | 2/a\|
2 | cot |-||
cot (a) \ \2//
----------- - --------------
1 2
1 + ------- / 1 \
2 |1 + -------|
cot (a) | 2/a\|
| cot |-||
\ \2//
$$- \frac{\left(1 - \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right)^{2}} + \frac{1 - \frac{1}{\cot^{2}{\left(a \right)}}}{1 + \frac{1}{\cot^{2}{\left(a \right)}}}$$
// 1 for a mod 2*pi = 0\ || | // 1 for a mod pi = 0\ - |< 2 | + |< | ||cos (a) otherwise | \\cos(2*a) otherwise / \\ /
$$\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\ || | // 1 for a mod pi = 0\ || 1 | || | - |<------- otherwise | + |< 1 | || 2 | ||-------- otherwise | ||sec (a) | \\sec(2*a) / \\ /
$$\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sec{\left(2 a \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\sec^{2}{\left(a \right)}} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\ // 1 for a mod pi = 0\ || | || | - |< 2/ pi\ | + |< /pi \ | ||sin |a + --| otherwise | ||sin|-- + 2*a| otherwise | \\ \ 2 / / \\ \2 / /
$$\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(2 a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\sin^{2}{\left(a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\ // 1 for a mod pi = 0\ || | || | || 1 | || 1 | - |<------------ otherwise | + |<------------- otherwise | || 2/pi \ | || /pi \ | ||csc |-- - a| | ||csc|-- - 2*a| | \\ \2 / / \\ \2 / /
$$\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
2
/ 4/a\\
4 | 4*sin |-||
4*sin (a) | \2/|
1 - --------- |1 - ---------|
2 | 2 |
sin (2*a) \ sin (a) /
------------- - ----------------
4 2
4*sin (a) / 4/a\\
1 + --------- | 4*sin |-||
2 | \2/|
sin (2*a) |1 + ---------|
| 2 |
\ sin (a) /
$$- \frac{\left(- \frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right)^{2}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right)^{2}} + \frac{- \frac{4 \sin^{4}{\left(a \right)}}{\sin^{2}{\left(2 a \right)}} + 1}{\frac{4 \sin^{4}{\left(a \right)}}{\sin^{2}{\left(2 a \right)}} + 1}$$
// / pi\ \ || 0 for |a + --| mod pi = 0| // /pi \ \ || \ 2 / | || 0 for |-- + 2*a| mod pi = 0| - |< | + |< \2 / | || 2 2/a pi\ | || | ||(1 + sin(a)) *cot |- + --| otherwise | \\cos(2*a) otherwise / \\ \2 4 / /
$$\left(- \begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(a \right)} + 1\right)^{2} \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(2 a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\ || | || 2 | // 1 for a mod pi = 0\ ||/ 2/a\\ | || | |||-1 + cot |-|| | || 2 | - |<\ \2// | + |<-1 + cot (a) | ||--------------- otherwise | ||------------ otherwise | || 2 | || 2 | || / 2/a\\ | \\1 + cot (a) / || |1 + cot |-|| | \\ \ \2// /
$$\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\ || | || 2 | // 1 for a mod pi = 0\ ||/ 2/a\\ | || | |||1 - tan |-|| | || 2 | - |<\ \2// | + |<1 - tan (a) | ||-------------- otherwise | ||----------- otherwise | || 2 | || 2 | ||/ 2/a\\ | \\1 + tan (a) / |||1 + tan |-|| | \\\ \2// /
$$\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{- \tan^{2}{\left(a \right)} + 1}{\tan^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\ || | || 2 | ||/ 2 4/a\\ | |||sin (a) - 4*sin |-|| | // 1 for a mod pi = 0\ - |<\ \2// | + |< | ||---------------------- otherwise | \\cos(2*a) otherwise / || 2 | ||/ 2 4/a\\ | |||sin (a) + 4*sin |-|| | \\\ \2// /
$$\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(- 4 \sin^{4}{\left(\frac{a}{2} \right)} + \sin^{2}{\left(a \right)}\right)^{2}}{\left(4 \sin^{4}{\left(\frac{a}{2} \right)} + \sin^{2}{\left(a \right)}\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\
|| | // 1 for a mod pi = 0\
||/ 1 for a mod 2*pi = 0 | || |
- |<| | + |
$$\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos^{2}{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
2
/ 2/a pi\\
2/ pi\ | cos |- - --||
cos |a - --| | \2 2 /|
\ 2 / |1 - ------------|
1 - ------------ | 2/a\ |
2 | cos |-| |
cos (a) \ \2/ /
---------------- - -------------------
2/ pi\ 2
cos |a - --| / 2/a pi\\
\ 2 / | cos |- - --||
1 + ------------ | \2 2 /|
2 |1 + ------------|
cos (a) | 2/a\ |
| cos |-| |
\ \2/ /
$$- \frac{\left(1 - \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right)^{2}}{\left(1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right)^{2}} + \frac{1 - \frac{\cos^{2}{\left(a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(a \right)}}}{1 + \frac{\cos^{2}{\left(a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(a \right)}}}$$
2
/ 2/a\ \
2 | sec |-| |
sec (a) | \2/ |
1 - ------------ |1 - ------------|
2/ pi\ | 2/a pi\|
sec |a - --| | sec |- - --||
\ 2 / \ \2 2 //
---------------- - -------------------
2 2
sec (a) / 2/a\ \
1 + ------------ | sec |-| |
2/ pi\ | \2/ |
sec |a - --| |1 + ------------|
\ 2 / | 2/a pi\|
| sec |- - --||
\ \2 2 //
$$- \frac{\left(- \frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}}{\left(\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}} + \frac{- \frac{\sec^{2}{\left(a \right)}}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} + 1}{\frac{\sec^{2}{\left(a \right)}}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} + 1}$$
2
/ 2/pi a\\
2/pi \ | csc |-- - -||
csc |-- - a| | \2 2/|
\2 / |1 - ------------|
1 - ------------ | 2/a\ |
2 | csc |-| |
csc (a) \ \2/ /
---------------- - -------------------
2/pi \ 2
csc |-- - a| / 2/pi a\\
\2 / | csc |-- - -||
1 + ------------ | \2 2/|
2 |1 + ------------|
csc (a) | 2/a\ |
| csc |-| |
\ \2/ /
$$- \frac{\left(1 - \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right)^{2}}{\left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right)^{2}} + \frac{1 - \frac{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(a \right)}}}{1 + \frac{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(a \right)}}}$$
// 1 for a mod 2*pi = 0\ || | || 2 | // 1 for a mod pi = 0\ ||/ 1 \ | || | |||-1 + -------| | || 1 | ||| 2/a\| | ||-1 + ------- | ||| tan |-|| | || 2 | - |<\ \2// | + |< tan (a) | ||--------------- otherwise | ||------------ otherwise | || 2 | || 1 | || / 1 \ | ||1 + ------- | || |1 + -------| | || 2 | || | 2/a\| | \\ tan (a) / || | tan |-|| | \\ \ \2// /
$$\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(a \right)}}}{1 + \frac{1}{\tan^{2}{\left(a \right)}}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// / pi\ \ || 0 for |a + --| mod pi = 0| // /pi \ \ || \ 2 / | || 0 for |-- + 2*a| mod pi = 0| || | || \2 / | || 2/a pi\ | || | || 4*cot |- + --| | || / pi\ | - |< \2 4 / | + |< 2*cot|a + --| | ||------------------- otherwise | || \ 4 / | || 2 | ||---------------- otherwise | ||/ 2/a pi\\ | || 2/ pi\ | |||1 + cot |- + --|| | ||1 + cot |a + --| | ||\ \2 4 // | \\ \ 4 / / \\ /
$$\left(- \begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(2 a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(a + \frac{\pi}{4} \right)}}{\cot^{2}{\left(a + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\ || | || 2 | // 1 for a mod pi = 0\ ||/ 2 \ | || | ||| sin (a) | | || 2 | |||-1 + ---------| | || sin (2*a) | ||| 4/a\| | ||-1 + --------- | ||| 4*sin |-|| | || 4 | - |<\ \2// | + |< 4*sin (a) | ||----------------- otherwise | ||-------------- otherwise | || 2 | || 2 | || / 2 \ | || sin (2*a) | || | sin (a) | | ||1 + --------- | || |1 + ---------| | || 4 | || | 4/a\| | \\ 4*sin (a) / || | 4*sin |-|| | \\ \ \2// /
$$\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(2 a \right)}}{4 \sin^{4}{\left(a \right)}}}{1 + \frac{\sin^{2}{\left(2 a \right)}}{4 \sin^{4}{\left(a \right)}}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\ || | ||/ 1 for a mod 2*pi = 0 | // 1 for a mod pi = 0\ ||| | || | ||| 2 | ||/ 1 for a mod pi = 0 | |||/ 2/a\\ | ||| | - |<||-1 + cot |-|| | + |<| 2 | ||<\ \2// otherwise | ||<-1 + cot (a) otherwise | |||--------------- otherwise | |||------------ otherwise | ||| 2 | ||| 2 | ||| / 2/a\\ | \\\1 + cot (a) / ||| |1 + cot |-|| | \\\ \ \2// /
$$\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\ || | || 2 | // 1 for a mod pi = 0\ ||/ 2/a\ \ | || | ||| cos |-| | | || 2 | ||| \2/ | | || cos (a) | |||-1 + ------------| | ||-1 + ------------ | ||| 2/a pi\| | || 2/ pi\ | ||| cos |- - --|| | || cos |a - --| | - |<\ \2 2 // | + |< \ 2 / | ||-------------------- otherwise | ||----------------- otherwise | || 2 | || 2 | ||/ 2/a\ \ | || cos (a) | ||| cos |-| | | || 1 + ------------ | ||| \2/ | | || 2/ pi\ | |||1 + ------------| | || cos |a - --| | ||| 2/a pi\| | \\ \ 2 / / ||| cos |- - --|| | \\\ \2 2 // /
$$\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\frac{\cos^{2}{\left(a \right)}}{\cos^{2}{\left(a - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(a \right)}}{\cos^{2}{\left(a - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\ || | || 2 | // 1 for a mod pi = 0\ ||/ 2/a pi\\ | || | ||| sec |- - --|| | || 2/ pi\ | ||| \2 2 /| | || sec |a - --| | |||-1 + ------------| | || \ 2 / | ||| 2/a\ | | ||-1 + ------------ | ||| sec |-| | | || 2 | - |<\ \2/ / | + |< sec (a) | ||-------------------- otherwise | ||----------------- otherwise | || 2 | || 2/ pi\ | ||/ 2/a pi\\ | || sec |a - --| | ||| sec |- - --|| | || \ 2 / | ||| \2 2 /| | || 1 + ------------ | |||1 + ------------| | || 2 | ||| 2/a\ | | \\ sec (a) / ||| sec |-| | | \\\ \2/ / /
$$\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(a \right)}}}{1 + \frac{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(a \right)}}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\ || | || 2 | // 1 for a mod pi = 0\ ||/ 2/a\ \ | || | ||| csc |-| | | || 2 | ||| \2/ | | || csc (a) | |||-1 + ------------| | ||-1 + ------------ | ||| 2/pi a\| | || 2/pi \ | ||| csc |-- - -|| | || csc |-- - a| | - |<\ \2 2// | + |< \2 / | ||-------------------- otherwise | ||----------------- otherwise | || 2 | || 2 | ||/ 2/a\ \ | || csc (a) | ||| csc |-| | | || 1 + ------------ | ||| \2/ | | || 2/pi \ | |||1 + ------------| | || csc |-- - a| | ||| 2/pi a\| | \\ \2 / / ||| csc |-- - -|| | \\\ \2 2// /
$$\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\frac{\csc^{2}{\left(a \right)}}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(a \right)}}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)$$
-Piecewise((1, Mod(a = 2*pi, 0)), ((-1 + csc(a/2)^2/csc(pi/2 - a/2)^2)^2/(1 + csc(a/2)^2/csc(pi/2 - a/2)^2)^2, True)) + Piecewise((1, Mod(a = pi, 0)), ((-1 + csc(a)^2/csc(pi/2 - a)^2)/(1 + csc(a)^2/csc(pi/2 - a)^2), True))