Господин Экзамен
Другие калькуляторы
-
Как пользоваться?
- cos(b)^2-cos(b)^2*sin(b)^2 если b=1/3
- 5*sin(a)^2+5*cos(a)^2+3 если a=-1/2
- sin(2*pi+a)+cos(pi+a)-sin(-a)+cos(-a) если a=1/3
- b/(b-4*sqrt(b))-4*b^(3/2)/(b^2-16*b) если b=1
- Разложить многочлен на множители:
- z^d+3*d+15*z^n+45*n
- -16+49*y^2
- 22*x*y^2+33*x^2*y
- n^2-20*n-100
- Общий знаменатель:
- a-c/c-a-c/a+c
- ((sin(4*a)/(1+cos(4*a))))*((cos(2*a)/(1+cos(2*a))))
- t^2/4*a+t-16*a^2/4*a+t
- (sin((13*pi)/2-x)-cot(6*pi+x))/(1+sin(2*pi-x))
- Умножение столбиком:
- 16*35
- 76*53
- 98*49
- 36*45
- Деление столбиком:
- 52068/2
- 67800/5
- 2164/6
- 5256/7
- Раскрыть скобки в:
- (1-x)*(1-x)
- (-x-4)*(-x-4)
- a*(b-c)-b*(a-c)-c*(b-a)
- 5*b^2*c^3-2*b*c^2*k-5*c*k^2+2*k^2
- Разложение числа на простые множители:
- 4365
- 38250
- 22120
- 264
- Производная:
- 1/3
- График функции y =:
- 1/3
- Интеграл d{x}:
-
1/3
-
Идентичные выражения
- cos(b)^ два -cos(b)^ два *sin(b)^ два если b= один / три
- косинус от (b) в квадрате минус косинус от (b) в квадрате умножить на синус от (b) в квадрате если b равно 1 делить на 3
- косинус от (b) в степени два минус косинус от (b) в степени два умножить на синус от (b) в степени два если b равно один делить на три
- cos(b)2-cos(b)2*sin(b)2 если b=1/3
- cosb2-cosb2*sinb2 если b=1/3
- cos(b)²-cos(b)²*sin(b)² если b=1/3
- cos(b) в степени 2-cos(b) в степени 2*sin(b) в степени 2 если b=1/3
- cos(b)^2-cos(b)^2sin(b)^2 если b=1/3
- cos(b)2-cos(b)2sin(b)2 если b=1/3
- cosb2-cosb2sinb2 если b=1/3
- cosb^2-cosb^2sinb^2 если b=1/3
- cos(b)^2-cos(b)^2*sin(b)^2 при b=1/3
- cos(b)^2-cos(b)^2*sin(b)^2 если b=1 разделить на 3
-
Похожие выражения
- cos(b)^2+cos(b)^2*sin(b)^2 если b=1/3
cos(b)^2-cos(b)^2*sin(b)^2 если b=1/3
Решение
Вы ввели
[src]
2 2 2 cos (b) - cos (b)*sin (b)
$$- \sin^{2}{\left(b \right)} \cos^{2}{\left(b \right)} + \cos^{2}{\left(b \right)}$$
cos(b)^2 - cos(b)^2*sin(b)^2
Подстановка условия
[src]
cos(b)^2 - cos(b)^2*sin(b)^2 при b = 1/3
подставляем
2 2 2 cos (b) - cos (b)*sin (b)
$$- \sin^{2}{\left(b \right)} \cos^{2}{\left(b \right)} + \cos^{2}{\left(b \right)}$$
4 cos (b)
$$\cos^{4}{\left(b \right)}$$
переменные
b = 1/3
$$b = \frac{1}{3}$$
4 cos ((1/3))
$$\cos^{4}{\left((1/3) \right)}$$
4 cos (1/3)
$$\cos^{4}{\left(\frac{1}{3} \right)}$$
cos(1/3)^4
Собрать выражение
[src]
3 cos(2*b) cos(4*b) - + -------- + -------- 8 2 8
$$\frac{\cos{\left(2 b \right)}}{2} + \frac{\cos{\left(4 b \right)}}{8} + \frac{3}{8}$$
3/8 + cos(2*b)/2 + cos(4*b)/8
Объединение рациональных выражений
[src]
2 / 2 \ cos (b)*\1 - sin (b)/
$$\left(- \sin^{2}{\left(b \right)} + 1\right) \cos^{2}{\left(b \right)}$$
cos(b)^2*(1 - sin(b)^2)
Степени
[src]
2
/ I*b -I*b\ 2
2 |e e | / -I*b I*b\
/ I*b -I*b\ |---- + -----| *\- e + e /
|e e | \ 2 2 /
|---- + -----| + ---------------------------------
\ 2 2 / 4
$$\frac{\left(\frac{e^{i b}}{2} + \frac{e^{- i b}}{2}\right)^{2} \left(e^{i b} - e^{- i b}\right)^{2}}{4} + \left(\frac{e^{i b}}{2} + \frac{e^{- i b}}{2}\right)^{2}$$
(exp(i*b)/2 + exp(-i*b)/2)^2 + (exp(i*b)/2 + exp(-i*b)/2)^2*(-exp(-i*b) + exp(i*b))^2/4
Комбинаторика
[src]
2 -cos (b)*(1 + sin(b))*(-1 + sin(b))
$$- \left(\sin{\left(b \right)} - 1\right) \left(\sin{\left(b \right)} + 1\right) \cos^{2}{\left(b \right)}$$
-cos(b)^2*(1 + sin(b))*(-1 + sin(b))
Тригонометрическая часть
[src]
4 cos (b)
$$\cos^{4}{\left(b \right)}$$
1 ------- 4 sec (b)
$$\frac{1}{\sec^{4}{\left(b \right)}}$$
4/ pi\
sin |b + --|
\ 2 /
$$\sin^{4}{\left(b + \frac{\pi}{2} \right)}$$
1
------------
4/pi \
csc |-- - b|
\2 /
$$\frac{1}{\csc^{4}{\left(- b + \frac{\pi}{2} \right)}}$$
3 cos(2*b) cos(4*b) - + -------- + -------- 8 2 8
$$\frac{\cos{\left(2 b \right)}}{2} + \frac{\cos{\left(4 b \right)}}{8} + \frac{3}{8}$$
4 / 2/b\\ 8/b\ |1 - tan |-|| *cos |-| \ \2// \2/
$$\left(- \tan^{2}{\left(\frac{b}{2} \right)} + 1\right)^{4} \cos^{8}{\left(\frac{b}{2} \right)}$$
2 2 2/ pi\
cos (b) - cos (b)*cos |b - --|
\ 2 /
$$- \cos^{2}{\left(b \right)} \cos^{2}{\left(b - \frac{\pi}{2} \right)} + \cos^{2}{\left(b \right)}$$
1 1 ------- - --------------- 2 2 2 sec (b) csc (b)*sec (b)
$$\frac{1}{\sec^{2}{\left(b \right)}} - \frac{1}{\csc^{2}{\left(b \right)} \sec^{2}{\left(b \right)}}$$
2/ pi\ 2 2/ pi\
sin |b + --| - sin (b)*sin |b + --|
\ 2 / \ 2 /
$$- \sin^{2}{\left(b \right)} \sin^{2}{\left(b + \frac{\pi}{2} \right)} + \sin^{2}{\left(b + \frac{\pi}{2} \right)}$$
1 1
------- - --------------------
2 2 2/ pi\
sec (b) sec (b)*sec |b - --|
\ 2 /
$$\frac{1}{\sec^{2}{\left(b \right)}} - \frac{1}{\sec^{2}{\left(b \right)} \sec^{2}{\left(b - \frac{\pi}{2} \right)}}$$
1 1
------- - --------------------
2 2 2/pi \
sec (b) sec (b)*sec |-- - b|
\2 /
$$\frac{1}{\sec^{2}{\left(b \right)}} - \frac{1}{\sec^{2}{\left(b \right)} \sec^{2}{\left(- b + \frac{\pi}{2} \right)}}$$
4
/ 2/b\\
|1 - tan |-||
\ \2//
--------------
4
/ 2/b\\
|1 + tan |-||
\ \2//
$$\frac{\left(- \tan^{2}{\left(\frac{b}{2} \right)} + 1\right)^{4}}{\left(\tan^{2}{\left(\frac{b}{2} \right)} + 1\right)^{4}}$$
1 1
------------ - --------------------
2/pi \ 2 2/pi \
csc |-- - b| csc (b)*csc |-- - b|
\2 / \2 /
$$\frac{1}{\csc^{2}{\left(- b + \frac{\pi}{2} \right)}} - \frac{1}{\csc^{2}{\left(b \right)} \csc^{2}{\left(- b + \frac{\pi}{2} \right)}}$$
1 1
------------ - -------------------------
2/pi \ 2 2/pi \
csc |-- - b| csc (pi - b)*csc |-- - b|
\2 / \2 /
$$\frac{1}{\csc^{2}{\left(- b + \frac{\pi}{2} \right)}} - \frac{1}{\csc^{2}{\left(- b + \pi \right)} \csc^{2}{\left(- b + \frac{\pi}{2} \right)}}$$
2 2 / /b\\ / /b\\ 2 4/b\ |1 + tan|-|| *|-1 + tan|-|| *cos (b)*cos |-| \ \2// \ \2// \2/
$$\left(\tan{\left(\frac{b}{2} \right)} - 1\right)^{2} \left(\tan{\left(\frac{b}{2} \right)} + 1\right)^{2} \cos^{4}{\left(\frac{b}{2} \right)} \cos^{2}{\left(b \right)}$$
2
1 + cos(2*b) / 2/b\\ 8/b\ 2/b\
------------ - 4*|1 - tan |-|| *cos |-|*tan |-|
2 \ \2// \2/ \2/
$$- 4 \left(- \tan^{2}{\left(\frac{b}{2} \right)} + 1\right)^{2} \cos^{8}{\left(\frac{b}{2} \right)} \tan^{2}{\left(\frac{b}{2} \right)} + \frac{\cos{\left(2 b \right)} + 1}{2}$$
/ 1 for b mod 2*pi = 0
|
| 4
$$\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{b}{2} \right)} - 1\right)^{4} \sin^{8}{\left(\frac{b}{2} \right)} & \text{otherwise} \end{cases}$$
1 cos(2*b) /1 cos(2*b)\ /1 cos(2*b)\ - + -------- - |- + --------|*|- - --------| 2 2 \2 2 / \2 2 /
$$- \left(- \frac{\cos{\left(2 b \right)}}{2} + \frac{1}{2}\right) \left(\frac{\cos{\left(2 b \right)}}{2} + \frac{1}{2}\right) + \frac{\cos{\left(2 b \right)}}{2} + \frac{1}{2}$$
2 1 cos(2*b) / 2/b\\ 6/b\ 2/b\ - + -------- - 4*|1 - tan |-|| *cos |-|*sin |-| 2 2 \ \2// \2/ \2/
$$- 4 \left(- \tan^{2}{\left(\frac{b}{2} \right)} + 1\right)^{2} \sin^{2}{\left(\frac{b}{2} \right)} \cos^{6}{\left(\frac{b}{2} \right)} + \frac{\cos{\left(2 b \right)}}{2} + \frac{1}{2}$$
/ 1 for b mod 2*pi = 0 | | 4 |/ 2/b\\ ||-1 + cot |-|| <\ \2// |--------------- otherwise | 4 | / 2/b\\ | |1 + cot |-|| \ \ \2//
$$\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{b}{2} \right)} - 1\right)^{4}}{\left(\cot^{2}{\left(\frac{b}{2} \right)} + 1\right)^{4}} & \text{otherwise} \end{cases}$$
2 2
1 + cos(2*b) / /b\\ / /b\\ 6/b\ 2/b\
------------ - 4*|1 + tan|-|| *|-1 + tan|-|| *cos |-|*sin |-|
2 \ \2// \ \2// \2/ \2/
$$- 4 \left(\tan{\left(\frac{b}{2} \right)} - 1\right)^{2} \left(\tan{\left(\frac{b}{2} \right)} + 1\right)^{2} \sin^{2}{\left(\frac{b}{2} \right)} \cos^{6}{\left(\frac{b}{2} \right)} + \frac{\cos{\left(2 b \right)} + 1}{2}$$
2 8 1 cos(2*b) / 2/b\\ / 2/b\\ 16/b\ 2/b\ - + -------- - 4*|1 - tan |-|| *|1 - tan |-|| *cos |-|*tan |-| 2 2 \ \2// \ \4// \4/ \2/
$$- 4 \left(- \tan^{2}{\left(\frac{b}{4} \right)} + 1\right)^{8} \left(- \tan^{2}{\left(\frac{b}{2} \right)} + 1\right)^{2} \cos^{16}{\left(\frac{b}{4} \right)} \tan^{2}{\left(\frac{b}{2} \right)} + \frac{\cos{\left(2 b \right)}}{2} + \frac{1}{2}$$
4 4
(1 - cos(b) + sin(b)) *(-1 + cos(b) + sin(b))
----------------------------------------------
4
/1 2 cos(2*b)\
|- + (1 - cos(b)) - --------|
\2 2 /
$$\frac{\left(\sin{\left(b \right)} - \cos{\left(b \right)} + 1\right)^{4} \left(\sin{\left(b \right)} + \cos{\left(b \right)} - 1\right)^{4}}{\left(\left(- \cos{\left(b \right)} + 1\right)^{2} - \frac{\cos{\left(2 b \right)}}{2} + \frac{1}{2}\right)^{4}}$$
2
/ 2/b pi\\
| cos |- - --||
1 cos(2*b) | \2 2 /| 6/b\ 2/b pi\
- + -------- - 4*|1 - ------------| *cos |-|*cos |- - --|
2 2 | 2/b\ | \2/ \2 2 /
| cos |-| |
\ \2/ /
$$- 4 \left(1 - \frac{\cos^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{b}{2} \right)}}\right)^{2} \cos^{6}{\left(\frac{b}{2} \right)} \cos^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)} + \frac{\cos{\left(2 b \right)}}{2} + \frac{1}{2}$$
2 2
/ 2/b pi\\ / 2/b\\ 2 4/b\
|1 - cot |- + --|| *|1 - tan |-|| *(1 + sin(b)) *cos |-|
1 + cos(2*b) \ \2 4 // \ \2// \2/
------------ - --------------------------------------------------------
2 4
$$- \frac{\left(- \tan^{2}{\left(\frac{b}{2} \right)} + 1\right)^{2} \left(- \cot^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\sin{\left(b \right)} + 1\right)^{2} \cos^{4}{\left(\frac{b}{2} \right)}}{4} + \frac{\cos{\left(2 b \right)} + 1}{2}$$
2
/ 2/b\ \
| sec |-| |
| \2/ |
4*|1 - ------------|
| 2/b pi\|
| sec |- - --||
1 1 \ \2 2 //
- + ---------- - ---------------------
2 2*sec(2*b) 6/b\ 2/b pi\
sec |-|*sec |- - --|
\2/ \2 2 /
$$\frac{1}{2} + \frac{1}{2 \sec{\left(2 b \right)}} - \frac{4 \left(- \frac{\sec^{2}{\left(\frac{b}{2} \right)}}{\sec^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}}{\sec^{6}{\left(\frac{b}{2} \right)} \sec^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}}$$
2
/ 4/b\\
| 4*sin |-||
| \2/| 4/b\ 8/pi b\
/pi \ 16*|1 - ---------| *sin |-|*sin |-- + -|
sin|-- + 2*b| | 2 | \2/ \2 2/
1 \2 / \ sin (b) /
- + ------------- - ----------------------------------------
2 2 2
sin (b)
$$- \frac{16 \left(- \frac{4 \sin^{4}{\left(\frac{b}{2} \right)}}{\sin^{2}{\left(b \right)}} + 1\right)^{2} \sin^{4}{\left(\frac{b}{2} \right)} \sin^{8}{\left(\frac{b}{2} + \frac{\pi}{2} \right)}}{\sin^{2}{\left(b \right)}} + \frac{\sin{\left(2 b + \frac{\pi}{2} \right)}}{2} + \frac{1}{2}$$
2 2
/ 2/b\\ / 2/b\\ 2/b\
|1 - tan |-|| 4*|1 - tan |-|| *tan |-|
\ \2// \ \2// \2/
-------------- - ------------------------
2 4
/ 2/b\\ / 2/b\\
|1 + tan |-|| |1 + tan |-||
\ \2// \ \2//
$$\frac{\left(- \tan^{2}{\left(\frac{b}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{b}{2} \right)} + 1\right)^{2}} - \frac{4 \left(- \tan^{2}{\left(\frac{b}{2} \right)} + 1\right)^{2} \tan^{2}{\left(\frac{b}{2} \right)}}{\left(\tan^{2}{\left(\frac{b}{2} \right)} + 1\right)^{4}}$$
2 2
/ 2/b\\ / 2/b pi\\ 4/b\
|-1 + cot |-|| *|-1 + tan |- + --|| *sin |-|
1 + cos(2*b) \ \2// \ \2 4 // \2/
------------ - --------------------------------------------
2 2
/ 2/b pi\\
|1 + tan |- + --||
\ \2 4 //
$$- \frac{\left(\tan^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)} - 1\right)^{2} \left(\cot^{2}{\left(\frac{b}{2} \right)} - 1\right)^{2} \sin^{4}{\left(\frac{b}{2} \right)}}{\left(\tan^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} + \frac{\cos{\left(2 b \right)} + 1}{2}$$
2
/ 2/pi b\\
| csc |-- - -||
| \2 2/|
4*|1 - ------------|
| 2/b\ |
| csc |-| |
1 1 \ \2/ /
- + --------------- - ---------------------
2 /pi \ 2/b\ 6/pi b\
2*csc|-- - 2*b| csc |-|*csc |-- - -|
\2 / \2/ \2 2/
$$\frac{1}{2} + \frac{1}{2 \csc{\left(- 2 b + \frac{\pi}{2} \right)}} - \frac{4 \left(1 - \frac{\csc^{2}{\left(- \frac{b}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{b}{2} \right)}}\right)^{2}}{\csc^{2}{\left(\frac{b}{2} \right)} \csc^{6}{\left(- \frac{b}{2} + \frac{\pi}{2} \right)}}$$
2 2 / 2 2 \ / 2 2 \ 1 cos (b) sin (b) |1 cos (b) sin (b)| |1 sin (b) cos (b)| - + ------- - ------- - |- + ------- - -------|*|- + ------- - -------| 2 2 2 \2 2 2 / \2 2 2 /
$$- \left(- \frac{\sin^{2}{\left(b \right)}}{2} + \frac{\cos^{2}{\left(b \right)}}{2} + \frac{1}{2}\right) \left(\frac{\sin^{2}{\left(b \right)}}{2} - \frac{\cos^{2}{\left(b \right)}}{2} + \frac{1}{2}\right) - \frac{\sin^{2}{\left(b \right)}}{2} + \frac{\cos^{2}{\left(b \right)}}{2} + \frac{1}{2}$$
2 8
/ 2/b\\ / 2/b\\ 2/b\
2 4*|1 - tan |-|| *|1 - tan |-|| *tan |-|
1 1 - tan (b) \ \2// \ \4// \2/
- + --------------- - ---------------------------------------
2 / 2 \ 8
2*\1 + tan (b)/ / 2/b\\
|1 + tan |-||
\ \4//
$$- \frac{4 \left(- \tan^{2}{\left(\frac{b}{4} \right)} + 1\right)^{8} \left(- \tan^{2}{\left(\frac{b}{2} \right)} + 1\right)^{2} \tan^{2}{\left(\frac{b}{2} \right)}}{\left(\tan^{2}{\left(\frac{b}{4} \right)} + 1\right)^{8}} + \frac{- \tan^{2}{\left(b \right)} + 1}{2 \left(\tan^{2}{\left(b \right)} + 1\right)} + \frac{1}{2}$$
2 2
/ 1 \ / 1 \
|1 - -------| 4*|1 - -------|
| 2/b\| | 2/b\|
| cot |-|| | cot |-||
\ \2// \ \2//
-------------- - ----------------------
2 4
/ 1 \ / 1 \ 2/b\
|1 + -------| |1 + -------| *cot |-|
| 2/b\| | 2/b\| \2/
| cot |-|| | cot |-||
\ \2// \ \2//
$$\frac{\left(1 - \frac{1}{\cot^{2}{\left(\frac{b}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{b}{2} \right)}}\right)^{2}} - \frac{4 \left(1 - \frac{1}{\cot^{2}{\left(\frac{b}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{b}{2} \right)}}\right)^{4} \cot^{2}{\left(\frac{b}{2} \right)}}$$
2/b pi\ 2/b\ 2/b pi\
4*tan |- + --| 16*cot |-|*tan |- + --|
\2 4 / \2/ \2 4 /
------------------- - ----------------------------------
2 2 2
/ 2/b pi\\ / 2/b\\ / 2/b pi\\
|1 + tan |- + --|| |1 + cot |-|| *|1 + tan |- + --||
\ \2 4 // \ \2// \ \2 4 //
$$\frac{4 \tan^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} - \frac{16 \tan^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)} \cot^{2}{\left(\frac{b}{2} \right)}}{\left(\tan^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\cot^{2}{\left(\frac{b}{2} \right)} + 1\right)^{2}}$$
2/b pi\ 2/b\ 2/b pi\
4*tan |- + --| 16*tan |-|*tan |- + --|
\2 4 / \2/ \2 4 /
------------------- - ----------------------------------
2 2 2
/ 2/b pi\\ / 2/b\\ / 2/b pi\\
|1 + tan |- + --|| |1 + tan |-|| *|1 + tan |- + --||
\ \2 4 // \ \2// \ \2 4 //
$$\frac{4 \tan^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} - \frac{16 \tan^{2}{\left(\frac{b}{2} \right)} \tan^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{b}{2} \right)} + 1\right)^{2} \left(\tan^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
// 0 for b mod pi = 0\ // 1 for b mod 2*pi = 0\ // 1 for b mod 2*pi = 0\ || | || | || | - |< 2 |*|< 2 | + |< 2 | ||sin (b) otherwise | ||cos (b) otherwise | ||cos (b) otherwise | \\ / \\ / \\ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\sin^{2}{\left(b \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\cos^{2}{\left(b \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\cos^{2}{\left(b \right)} & \text{otherwise} \end{cases}\right)$$
// 0 for b mod pi = 0\ // 1 for b mod 2*pi = 0\ // 1 for b mod 2*pi = 0\ || | || | || | - |< 2/ pi\ |*|< 2 | + |< 2 | ||cos |b - --| otherwise | ||cos (b) otherwise | ||cos (b) otherwise | \\ \ 2 / / \\ / \\ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\cos^{2}{\left(b - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\cos^{2}{\left(b \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\cos^{2}{\left(b \right)} & \text{otherwise} \end{cases}\right)$$
2 2 2
/ 2/b\\ / 2/b\\ / 2/b pi\\
|-1 + cot |-|| |-1 + cot |-|| *|-1 + tan |- + --||
\ \2// \ \2// \ \2 4 //
--------------- - ------------------------------------
2 2 2
/ 2/b\\ / 2/b\\ / 2/b pi\\
|1 + cot |-|| |1 + cot |-|| *|1 + tan |- + --||
\ \2// \ \2// \ \2 4 //
$$- \frac{\left(\tan^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)} - 1\right)^{2} \left(\cot^{2}{\left(\frac{b}{2} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\cot^{2}{\left(\frac{b}{2} \right)} + 1\right)^{2}} + \frac{\left(\cot^{2}{\left(\frac{b}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{b}{2} \right)} + 1\right)^{2}}$$
2 2 2
/ 2/b\\ / 2/b pi\\ / 2/b\\
|1 - tan |-|| |1 - cot |- + --|| *|1 - tan |-||
\ \2// \ \2 4 // \ \2//
-------------- - ----------------------------------
2 2 2
/ 2/b\\ / 2/b pi\\ / 2/b\\
|1 + tan |-|| |1 + cot |- + --|| *|1 + tan |-||
\ \2// \ \2 4 // \ \2//
$$- \frac{\left(- \tan^{2}{\left(\frac{b}{2} \right)} + 1\right)^{2} \left(- \cot^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{b}{2} \right)} + 1\right)^{2} \left(\cot^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} + \frac{\left(- \tan^{2}{\left(\frac{b}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{b}{2} \right)} + 1\right)^{2}}$$
// 0 for b mod pi = 0\ // 1 for b mod 2*pi = 0\ // 1 for b mod 2*pi = 0\ || | || | || | - |< 2 |*|< 2/ pi\ | + |< 2/ pi\ | ||sin (b) otherwise | ||sin |b + --| otherwise | ||sin |b + --| otherwise | \\ / \\ \ 2 / / \\ \ 2 / /
$$\left(- \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\sin^{2}{\left(b \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\sin^{2}{\left(b + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\sin^{2}{\left(b + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)$$
// 0 for b mod pi = 0\ // 1 for b mod 2*pi = 0\ // 1 for b mod 2*pi = 0\ || | || | || | || 1 | || 1 | || 1 | - |<------------ otherwise |*|<------- otherwise | + |<------- otherwise | || 2/ pi\ | || 2 | || 2 | ||sec |b - --| | ||sec (b) | ||sec (b) | \\ \ 2 / / \\ / \\ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\frac{1}{\sec^{2}{\left(b - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{1}{\sec^{2}{\left(b \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{1}{\sec^{2}{\left(b \right)}} & \text{otherwise} \end{cases}\right)$$
// 0 for b mod pi = 0\ // 1 for b mod 2*pi = 0\ // 1 for b mod 2*pi = 0\ || | || | || | || 1 | || 1 | || 1 | - |<------- otherwise |*|<------------ otherwise | + |<------------ otherwise | || 2 | || 2/pi \ | || 2/pi \ | ||csc (b) | ||csc |-- - b| | ||csc |-- - b| | \\ / \\ \2 / / \\ \2 / /
$$\left(- \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\frac{1}{\csc^{2}{\left(b \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{1}{\csc^{2}{\left(- b + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{1}{\csc^{2}{\left(- b + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// b \
/ 1 for b mod pi = 0 || 1 for - mod 2*pi = 0|
< 2 || 2 |
1 \cos(2*b) otherwise / 2/b\\ 2/b\ || |
- + --------------------------- - 4*|1 - tan |-|| *tan |-|*|< 8 |
2 2 \ \2// \2/ ||/ 2/b\\ 16/b\ |
|||-1 + cot |-|| *sin |-| otherwise |
||\ \4// \4/ |
\\ /
$$\left(- 4 \left(- \tan^{2}{\left(\frac{b}{2} \right)} + 1\right)^{2} \left(\begin{cases} 1 & \text{for}\: \frac{b}{2} \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{b}{4} \right)} - 1\right)^{8} \sin^{16}{\left(\frac{b}{4} \right)} & \text{otherwise} \end{cases}\right) \tan^{2}{\left(\frac{b}{2} \right)}\right) + \left(\frac{\begin{cases} 1 & \text{for}\: b \bmod \pi = 0 \\\cos{\left(2 b \right)} & \text{otherwise} \end{cases}}{2}\right) + \frac{1}{2}$$
2 2
/ 4/b\\ / 4/b\\
| 4*sin |-|| | 4*sin |-||
| \2/| | \2/| 4/b\
|1 - ---------| 16*|1 - ---------| *sin |-|
| 2 | | 2 | \2/
\ sin (b) / \ sin (b) /
---------------- - ---------------------------
2 4
/ 4/b\\ / 4/b\\
| 4*sin |-|| | 4*sin |-||
| \2/| | \2/| 2
|1 + ---------| |1 + ---------| *sin (b)
| 2 | | 2 |
\ sin (b) / \ sin (b) /
$$\frac{\left(- \frac{4 \sin^{4}{\left(\frac{b}{2} \right)}}{\sin^{2}{\left(b \right)}} + 1\right)^{2}}{\left(\frac{4 \sin^{4}{\left(\frac{b}{2} \right)}}{\sin^{2}{\left(b \right)}} + 1\right)^{2}} - \frac{16 \left(- \frac{4 \sin^{4}{\left(\frac{b}{2} \right)}}{\sin^{2}{\left(b \right)}} + 1\right)^{2} \sin^{4}{\left(\frac{b}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{b}{2} \right)}}{\sin^{2}{\left(b \right)}} + 1\right)^{4} \sin^{2}{\left(b \right)}}$$
// / 3*pi\ \
// 1 for b mod 2*pi = 0\ || 1 for |b + ----| mod 2*pi = 0| // 1 for b mod 2*pi = 0\
|| | || \ 2 / | || |
- |< 2 |*|< | + |< 2 |
||cos (b) otherwise | || 4/b\ 2/b\ | ||cos (b) otherwise |
\\ / ||- 4*cos |-| + 4*cos |-| otherwise | \\ /
\\ \2/ \2/ /
$$\left(- \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\cos^{2}{\left(b \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(b + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\- 4 \cos^{4}{\left(\frac{b}{2} \right)} + 4 \cos^{2}{\left(\frac{b}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\cos^{2}{\left(b \right)} & \text{otherwise} \end{cases}\right)$$
// b \
|| 1 for - mod 2*pi = 0|
|| 2 |
|| |
2 || 8 |
/ 1 \ ||/ 2/b\\ |
/ 1 for b mod pi = 0 4*|1 - -------| *|<|-1 + cot |-|| |
| | 2/b\| ||\ \4// |
| 2 | cot |-|| ||--------------- otherwise |
<-1 + cot (b) \ \2// || 8 |
|------------ otherwise || / 2/b\\ |
| 2 || |1 + cot |-|| |
1 \1 + cot (b) \\ \ \4// /
- + ------------------------------- - -------------------------------------------------------
2 2 2/b\
cot |-|
\2/
$$\left(- \frac{4 \left(1 - \frac{1}{\cot^{2}{\left(\frac{b}{2} \right)}}\right)^{2} \left(\begin{cases} 1 & \text{for}\: \frac{b}{2} \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{b}{4} \right)} - 1\right)^{8}}{\left(\cot^{2}{\left(\frac{b}{4} \right)} + 1\right)^{8}} & \text{otherwise} \end{cases}\right)}{\cot^{2}{\left(\frac{b}{2} \right)}}\right) + \left(\frac{\begin{cases} 1 & \text{for}\: b \bmod \pi = 0 \\\frac{\cot^{2}{\left(b \right)} - 1}{\cot^{2}{\left(b \right)} + 1} & \text{otherwise} \end{cases}}{2}\right) + \frac{1}{2}$$
// / pi\ \ // / pi\ \
// 0 for b mod pi = 0\ || 0 for |b + --| mod pi = 0| || 0 for |b + --| mod pi = 0|
|| | || \ 2 / | || \ 2 / |
- |< 2 |*|< | + |< |
||sin (b) otherwise | || 2 2/b pi\ | || 2 2/b pi\ |
\\ / ||(1 + sin(b)) *cot |- + --| otherwise | ||(1 + sin(b)) *cot |- + --| otherwise |
\\ \2 4 / / \\ \2 4 / /
$$\left(- \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\sin^{2}{\left(b \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(b + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(b \right)} + 1\right)^{2} \cot^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \left(b + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(b \right)} + 1\right)^{2} \cot^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)$$
2 2
/ 2/b pi\\ / 2/b pi\\
| cos |- - --|| | cos |- - --||
| \2 2 /| | \2 2 /| 2/b pi\
|1 - ------------| 4*|1 - ------------| *cos |- - --|
| 2/b\ | | 2/b\ | \2 2 /
| cos |-| | | cos |-| |
\ \2/ / \ \2/ /
------------------- - ----------------------------------
2 4
/ 2/b pi\\ / 2/b pi\\
| cos |- - --|| | cos |- - --||
| \2 2 /| | \2 2 /| 2/b\
|1 + ------------| |1 + ------------| *cos |-|
| 2/b\ | | 2/b\ | \2/
| cos |-| | | cos |-| |
\ \2/ / \ \2/ /
$$\frac{\left(1 - \frac{\cos^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{b}{2} \right)}}\right)^{2}}{\left(1 + \frac{\cos^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{b}{2} \right)}}\right)^{2}} - \frac{4 \left(1 - \frac{\cos^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{b}{2} \right)}}\right)^{2} \cos^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{b}{2} \right)}}\right)^{4} \cos^{2}{\left(\frac{b}{2} \right)}}$$
2 2
/ 2/b\ \ / 2/b\ \
| sec |-| | | sec |-| |
| \2/ | | \2/ | 2/b\
|1 - ------------| 4*|1 - ------------| *sec |-|
| 2/b pi\| | 2/b pi\| \2/
| sec |- - --|| | sec |- - --||
\ \2 2 // \ \2 2 //
------------------- - --------------------------------
2 4
/ 2/b\ \ / 2/b\ \
| sec |-| | | sec |-| |
| \2/ | | \2/ | 2/b pi\
|1 + ------------| |1 + ------------| *sec |- - --|
| 2/b pi\| | 2/b pi\| \2 2 /
| sec |- - --|| | sec |- - --||
\ \2 2 // \ \2 2 //
$$\frac{\left(- \frac{\sec^{2}{\left(\frac{b}{2} \right)}}{\sec^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}}{\left(\frac{\sec^{2}{\left(\frac{b}{2} \right)}}{\sec^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}} - \frac{4 \left(- \frac{\sec^{2}{\left(\frac{b}{2} \right)}}{\sec^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec^{2}{\left(\frac{b}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{b}{2} \right)}}{\sec^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}} + 1\right)^{4} \sec^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}}$$
2 2
/ 2/pi b\\ / 2/pi b\\
| csc |-- - -|| | csc |-- - -||
| \2 2/| | \2 2/| 2/pi b\
|1 - ------------| 4*|1 - ------------| *csc |-- - -|
| 2/b\ | | 2/b\ | \2 2/
| csc |-| | | csc |-| |
\ \2/ / \ \2/ /
------------------- - ----------------------------------
2 4
/ 2/pi b\\ / 2/pi b\\
| csc |-- - -|| | csc |-- - -||
| \2 2/| | \2 2/| 2/b\
|1 + ------------| |1 + ------------| *csc |-|
| 2/b\ | | 2/b\ | \2/
| csc |-| | | csc |-| |
\ \2/ / \ \2/ /
$$\frac{\left(1 - \frac{\csc^{2}{\left(- \frac{b}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{b}{2} \right)}}\right)^{2}}{\left(1 + \frac{\csc^{2}{\left(- \frac{b}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{b}{2} \right)}}\right)^{2}} - \frac{4 \left(1 - \frac{\csc^{2}{\left(- \frac{b}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{b}{2} \right)}}\right)^{2} \csc^{2}{\left(- \frac{b}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{b}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{b}{2} \right)}}\right)^{4} \csc^{2}{\left(\frac{b}{2} \right)}}$$
// 0 for b mod pi = 0\ // 1 for b mod 2*pi = 0\ // 1 for b mod 2*pi = 0\ || | || | || | ||/ 0 for b mod pi = 0 | ||/ 1 for b mod 2*pi = 0 | ||/ 1 for b mod 2*pi = 0 | - |<| |*|<| | + |<| | ||< 2 otherwise | ||< 2 otherwise | ||< 2 otherwise | |||sin (b) otherwise | |||cos (b) otherwise | |||cos (b) otherwise | \\\ / \\\ / \\\ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\sin^{2}{\left(b \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\cos^{2}{\left(b \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\cos^{2}{\left(b \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
// 0 for b mod pi = 0\ // 1 for b mod 2*pi = 0\ // 1 for b mod 2*pi = 0\ || | || | || | || 2/b\ | || 2 | || 2 | || 4*cot |-| | ||/ 2/b\\ | ||/ 2/b\\ | || \2/ | |||-1 + cot |-|| | |||-1 + cot |-|| | - |<-------------- otherwise |*|<\ \2// | + |<\ \2// | || 2 | ||--------------- otherwise | ||--------------- otherwise | ||/ 2/b\\ | || 2 | || 2 | |||1 + cot |-|| | || / 2/b\\ | || / 2/b\\ | ||\ \2// | || |1 + cot |-|| | || |1 + cot |-|| | \\ / \\ \ \2// / \\ \ \2// /
$$\left(- \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{b}{2} \right)}}{\left(\cot^{2}{\left(\frac{b}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{b}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{b}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{b}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{b}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// 0 for b mod pi = 0\ // 1 for b mod 2*pi = 0\ // 1 for b mod 2*pi = 0\ || | || | || | || 2/b\ | || 2 | || 2 | || 4*tan |-| | ||/ 2/b\\ | ||/ 2/b\\ | || \2/ | |||1 - tan |-|| | |||1 - tan |-|| | - |<-------------- otherwise |*|<\ \2// | + |<\ \2// | || 2 | ||-------------- otherwise | ||-------------- otherwise | ||/ 2/b\\ | || 2 | || 2 | |||1 + tan |-|| | ||/ 2/b\\ | ||/ 2/b\\ | ||\ \2// | |||1 + tan |-|| | |||1 + tan |-|| | \\ / \\\ \2// / \\\ \2// /
$$\left(- \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\frac{4 \tan^{2}{\left(\frac{b}{2} \right)}}{\left(\tan^{2}{\left(\frac{b}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{\left(- \tan^{2}{\left(\frac{b}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{b}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{\left(- \tan^{2}{\left(\frac{b}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{b}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// 1 for b mod 2*pi = 0\ // 1 for b mod 2*pi = 0\
|| | || |
// 0 for b mod pi = 0\ || 2 | || 2 |
|| | ||/ 1 \ | ||/ 1 \ |
|| 4 | |||-1 + -------| | |||-1 + -------| |
||---------------------- otherwise | ||| 2/b\| | ||| 2/b\| |
|| 2 | ||| tan |-|| | ||| tan |-|| |
- |
$$\left(- \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\frac{4}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{b}{2} \right)}}\right)^{2} \tan^{2}{\left(\frac{b}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{b}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{b}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{b}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{b}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// / pi\ \ // / pi\ \
// 0 for b mod pi = 0\ || 0 for |b + --| mod pi = 0| || 0 for |b + --| mod pi = 0|
|| | || \ 2 / | || \ 2 / |
|| 2/b\ | || | || |
|| 4*cot |-| | || 2/b pi\ | || 2/b pi\ |
|| \2/ | || 4*cot |- + --| | || 4*cot |- + --| |
- |<-------------- otherwise |*|< \2 4 / | + |< \2 4 / |
|| 2 | ||------------------- otherwise | ||------------------- otherwise |
||/ 2/b\\ | || 2 | || 2 |
|||1 + cot |-|| | ||/ 2/b pi\\ | ||/ 2/b pi\\ |
||\ \2// | |||1 + cot |- + --|| | |||1 + cot |- + --|| |
\\ / ||\ \2 4 // | ||\ \2 4 // |
\\ / \\ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{b}{2} \right)}}{\left(\cot^{2}{\left(\frac{b}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(b + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)}}{\left(\cot^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \left(b + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)}}{\left(\cot^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// / 3*pi\ \
// 1 for b mod 2*pi = 0\ || 1 for |b + ----| mod 2*pi = 0| // 1 for b mod 2*pi = 0\
|| | || \ 2 / | || |
|| 2 | || | || 2 |
||/ 2/b\\ | || 2 | ||/ 2/b\\ |
|||-1 + cot |-|| | ||/ 2/b pi\\ | |||-1 + cot |-|| |
- |<\ \2// |*|<|-1 + tan |- + --|| | + |<\ \2// |
||--------------- otherwise | ||\ \2 4 // | ||--------------- otherwise |
|| 2 | ||-------------------- otherwise | || 2 |
|| / 2/b\\ | || 2 | || / 2/b\\ |
|| |1 + cot |-|| | ||/ 2/b pi\\ | || |1 + cot |-|| |
\\ \ \2// / |||1 + tan |- + --|| | \\ \ \2// /
\\\ \2 4 // /
$$\left(- \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{b}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{b}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(b + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\left(\tan^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{b}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{b}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{b}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// 0 for b mod pi = 0\
|| | // 1 for b mod 2*pi = 0\ // 1 for b mod 2*pi = 0\
|| 2 | || | || |
|| sin (b) | || 2 | || 2 |
||------------------------ otherwise | ||/ 2 4/b\\ | ||/ 2 4/b\\ |
|| 2 | |||sin (b) - 4*sin |-|| | |||sin (b) - 4*sin |-|| |
- |
$$\left(- \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\frac{\sin^{2}{\left(b \right)}}{\left(1 + \frac{\sin^{2}{\left(b \right)}}{4 \sin^{4}{\left(\frac{b}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{b}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{\left(- 4 \sin^{4}{\left(\frac{b}{2} \right)} + \sin^{2}{\left(b \right)}\right)^{2}}{\left(4 \sin^{4}{\left(\frac{b}{2} \right)} + \sin^{2}{\left(b \right)}\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{\left(- 4 \sin^{4}{\left(\frac{b}{2} \right)} + \sin^{2}{\left(b \right)}\right)^{2}}{\left(4 \sin^{4}{\left(\frac{b}{2} \right)} + \sin^{2}{\left(b \right)}\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// 1 for b mod 2*pi = 0\ // 1 for b mod 2*pi = 0\
|| | || |
// 0 for b mod pi = 0\ || 2 | || 2 |
|| | ||/ 2 \ | ||/ 2 \ |
|| 2 | ||| sin (b) | | ||| sin (b) | |
|| sin (b) | |||-1 + ---------| | |||-1 + ---------| |
||------------------------ otherwise | ||| 4/b\| | ||| 4/b\| |
|| 2 | ||| 4*sin |-|| | ||| 4*sin |-|| |
- |
$$\left(- \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\frac{\sin^{2}{\left(b \right)}}{\left(1 + \frac{\sin^{2}{\left(b \right)}}{4 \sin^{4}{\left(\frac{b}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{b}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sin^{2}{\left(b \right)}}{4 \sin^{4}{\left(\frac{b}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sin^{2}{\left(b \right)}}{4 \sin^{4}{\left(\frac{b}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sin^{2}{\left(b \right)}}{4 \sin^{4}{\left(\frac{b}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sin^{2}{\left(b \right)}}{4 \sin^{4}{\left(\frac{b}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// 0 for b mod pi = 0\ // 1 for b mod 2*pi = 0\ // 1 for b mod 2*pi = 0\ || | || | || | ||/ 0 for b mod pi = 0 | ||/ 1 for b mod 2*pi = 0 | ||/ 1 for b mod 2*pi = 0 | ||| | ||| | ||| | ||| 2/b\ | ||| 2 | ||| 2 | ||| 4*cot |-| | |||/ 2/b\\ | |||/ 2/b\\ | - |<| \2/ |*|<||-1 + cot |-|| | + |<||-1 + cot |-|| | ||<-------------- otherwise otherwise | ||<\ \2// otherwise | ||<\ \2// otherwise | ||| 2 | |||--------------- otherwise | |||--------------- otherwise | |||/ 2/b\\ | ||| 2 | ||| 2 | ||||1 + cot |-|| | ||| / 2/b\\ | ||| / 2/b\\ | |||\ \2// | ||| |1 + cot |-|| | ||| |1 + cot |-|| | \\\ / \\\ \ \2// / \\\ \ \2// /
$$\left(- \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{b}{2} \right)}}{\left(\cot^{2}{\left(\frac{b}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{b}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{b}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{b}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{b}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
// 1 for b mod 2*pi = 0\ // 1 for b mod 2*pi = 0\
|| | || |
// 0 for b mod pi = 0\ || 2 | || 2 |
|| | ||/ 2/b\ \ | ||/ 2/b\ \ |
|| 2/b\ | ||| cos |-| | | ||| cos |-| | |
|| 4*cos |-| | ||| \2/ | | ||| \2/ | |
|| \2/ | |||-1 + ------------| | |||-1 + ------------| |
||-------------------------------- otherwise | ||| 2/b pi\| | ||| 2/b pi\| |
|| 2 | ||| cos |- - --|| | ||| cos |- - --|| |
- |
$$\left(- \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\frac{4 \cos^{2}{\left(\frac{b}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{b}{2} \right)}}{\cos^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \cos^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{\left(\frac{\cos^{2}{\left(\frac{b}{2} \right)}}{\cos^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\cos^{2}{\left(\frac{b}{2} \right)}}{\cos^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{\left(\frac{\cos^{2}{\left(\frac{b}{2} \right)}}{\cos^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\cos^{2}{\left(\frac{b}{2} \right)}}{\cos^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// 1 for b mod 2*pi = 0\ // 1 for b mod 2*pi = 0\
|| | || |
// 0 for b mod pi = 0\ || 2 | || 2 |
|| | ||/ 2/b pi\\ | ||/ 2/b pi\\ |
|| 2/b pi\ | ||| sec |- - --|| | ||| sec |- - --|| |
|| 4*sec |- - --| | ||| \2 2 /| | ||| \2 2 /| |
|| \2 2 / | |||-1 + ------------| | |||-1 + ------------| |
||--------------------------- otherwise | ||| 2/b\ | | ||| 2/b\ | |
|| 2 | ||| sec |-| | | ||| sec |-| | |
- |
$$\left(- \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\frac{4 \sec^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{b}{2} \right)}}\right)^{2} \sec^{2}{\left(\frac{b}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{b}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sec^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{b}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{b}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sec^{2}{\left(\frac{b}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{b}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// 1 for b mod 2*pi = 0\ // 1 for b mod 2*pi = 0\
|| | || |
// 0 for b mod pi = 0\ || 2 | || 2 |
|| | ||/ 2/b\ \ | ||/ 2/b\ \ |
|| 2/b\ | ||| csc |-| | | ||| csc |-| | |
|| 4*csc |-| | ||| \2/ | | ||| \2/ | |
|| \2/ | |||-1 + ------------| | |||-1 + ------------| |
||-------------------------------- otherwise | ||| 2/pi b\| | ||| 2/pi b\| |
|| 2 | ||| csc |-- - -|| | ||| csc |-- - -|| |
- |
$$\left(- \left(\begin{cases} 0 & \text{for}\: b \bmod \pi = 0 \\\frac{4 \csc^{2}{\left(\frac{b}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{b}{2} \right)}}{\csc^{2}{\left(- \frac{b}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2} \csc^{2}{\left(- \frac{b}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{b}{2} \right)}}{\csc^{2}{\left(- \frac{b}{2} + \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\csc^{2}{\left(\frac{b}{2} \right)}}{\csc^{2}{\left(- \frac{b}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: b \bmod 2 \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{b}{2} \right)}}{\csc^{2}{\left(- \frac{b}{2} + \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\csc^{2}{\left(\frac{b}{2} \right)}}{\csc^{2}{\left(- \frac{b}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)$$
-Piecewise((0, Mod(b = pi, 0)), (4*csc(b/2)^2/((1 + csc(b/2)^2/csc(pi/2 - b/2)^2)^2*csc(pi/2 - b/2)^2), True))*Piecewise((1, Mod(b = 2*pi, 0)), ((-1 + csc(b/2)^2/csc(pi/2 - b/2)^2)^2/(1 + csc(b/2)^2/csc(pi/2 - b/2)^2)^2, True)) + Piecewise((1, Mod(b = 2*pi, 0)), ((-1 + csc(b/2)^2/csc(pi/2 - b/2)^2)^2/(1 + csc(b/2)^2/csc(pi/2 - b/2)^2)^2, True))