Господин Экзамен
sin(p)^2-sin(p)^2*cos(p)^2 если p=3
Решение
Вы ввели
[src]
2 2 2 sin (p) - sin (p)*cos (p)
$$- \sin^{2}{\left(p \right)} \cos^{2}{\left(p \right)} + \sin^{2}{\left(p \right)}$$
sin(p)^2 - sin(p)^2*cos(p)^2
Подстановка условия
[src]
sin(p)^2 - sin(p)^2*cos(p)^2 при p = 3
подставляем
2 2 2 sin (p) - sin (p)*cos (p)
$$- \sin^{2}{\left(p \right)} \cos^{2}{\left(p \right)} + \sin^{2}{\left(p \right)}$$
4 sin (p)
$$\sin^{4}{\left(p \right)}$$
переменные
p = 3
$$p = 3$$
4 sin ((3))
$$\sin^{4}{\left((3) \right)}$$
4 sin (3)
$$\sin^{4}{\left(3 \right)}$$
sin(3)^4
Собрать выражение
[src]
3 cos(2*p) cos(4*p) - - -------- + -------- 8 2 8
$$- \frac{\cos{\left(2 p \right)}}{2} + \frac{\cos{\left(4 p \right)}}{8} + \frac{3}{8}$$
3/8 - cos(2*p)/2 + cos(4*p)/8
Тригонометрическая часть
[src]
4 sin (p)
$$\sin^{4}{\left(p \right)}$$
1 ------- 4 csc (p)
$$\frac{1}{\csc^{4}{\left(p \right)}}$$
4/ pi\
cos |p - --|
\ 2 /
$$\cos^{4}{\left(p - \frac{\pi}{2} \right)}$$
1
------------
4/ pi\
sec |p - --|
\ 2 /
$$\frac{1}{\sec^{4}{\left(p - \frac{\pi}{2} \right)}}$$
8/p\ 4/p\
16*cos |-|*tan |-|
\2/ \2/
$$16 \cos^{8}{\left(\frac{p}{2} \right)} \tan^{4}{\left(\frac{p}{2} \right)}$$
2 1 cos(2*p) sin (2*p) - - -------- - --------- 2 2 4
$$- \frac{\sin^{2}{\left(2 p \right)}}{4} - \frac{\cos{\left(2 p \right)}}{2} + \frac{1}{2}$$
2 2 2/ pi\
sin (p) - sin (p)*sin |p + --|
\ 2 /
$$- \sin^{2}{\left(p \right)} \sin^{2}{\left(p + \frac{\pi}{2} \right)} + \sin^{2}{\left(p \right)}$$
2/p\ 2 2/p\
4*cos |-|*sin (p)*sin |-|
\2/ \2/
$$4 \sin^{2}{\left(\frac{p}{2} \right)} \sin^{2}{\left(p \right)} \cos^{2}{\left(\frac{p}{2} \right)}$$
1 1 ------- - --------------- 2 2 2 csc (p) csc (p)*sec (p)
$$\frac{1}{\csc^{2}{\left(p \right)}} - \frac{1}{\csc^{2}{\left(p \right)} \sec^{2}{\left(p \right)}}$$
4/p\
16*tan |-|
\2/
--------------
4
/ 2/p\\
|1 + tan |-||
\ \2//
$$\frac{16 \tan^{4}{\left(\frac{p}{2} \right)}}{\left(\tan^{2}{\left(\frac{p}{2} \right)} + 1\right)^{4}}$$
2/ pi\ 2 2/ pi\
cos |p - --| - cos (p)*cos |p - --|
\ 2 / \ 2 /
$$- \cos^{2}{\left(p \right)} \cos^{2}{\left(p - \frac{\pi}{2} \right)} + \cos^{2}{\left(p - \frac{\pi}{2} \right)}$$
1 1
------- - --------------------
2 2 2/pi \
csc (p) csc (p)*csc |-- - p|
\2 /
$$\frac{1}{\csc^{2}{\left(p \right)}} - \frac{1}{\csc^{2}{\left(p \right)} \csc^{2}{\left(- p + \frac{\pi}{2} \right)}}$$
1 1
------------ - --------------------
2/ pi\ 2 2/ pi\
sec |p - --| sec (p)*sec |p - --|
\ 2 / \ 2 /
$$\frac{1}{\sec^{2}{\left(p - \frac{\pi}{2} \right)}} - \frac{1}{\sec^{2}{\left(p \right)} \sec^{2}{\left(p - \frac{\pi}{2} \right)}}$$
1 1
------------ - -------------------------
2 2 2/pi \
csc (pi - p) csc (pi - p)*csc |-- - p|
\2 /
$$\frac{1}{\csc^{2}{\left(- p + \pi \right)}} - \frac{1}{\csc^{2}{\left(- p + \pi \right)} \csc^{2}{\left(- p + \frac{\pi}{2} \right)}}$$
1 1
------------ - --------------------
2/pi \ 2 2/pi \
sec |-- - p| sec (p)*sec |-- - p|
\2 / \2 /
$$\frac{1}{\sec^{2}{\left(- p + \frac{\pi}{2} \right)}} - \frac{1}{\sec^{2}{\left(p \right)} \sec^{2}{\left(- p + \frac{\pi}{2} \right)}}$$
/ 0 for p mod pi = 0 | < 4/p\ 8/p\ |16*cot |-|*sin |-| otherwise \ \2/ \2/
$$\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\16 \sin^{8}{\left(\frac{p}{2} \right)} \cot^{4}{\left(\frac{p}{2} \right)} & \text{otherwise} \end{cases}$$
4 8/p\
256*sin (p)*sin |-|
\2/
----------------------
4
/ 2 4/p\\
|sin (p) + 4*sin |-||
\ \2//
$$\frac{256 \sin^{8}{\left(\frac{p}{2} \right)} \sin^{4}{\left(p \right)}}{\left(4 \sin^{4}{\left(\frac{p}{2} \right)} + \sin^{2}{\left(p \right)}\right)^{4}}$$
1 cos(2*p) /1 cos(2*p)\ /1 cos(2*p)\ - - -------- - |- + --------|*|- - --------| 2 2 \2 2 / \2 2 /
$$- \left(- \frac{\cos{\left(2 p \right)}}{2} + \frac{1}{2}\right) \left(\frac{\cos{\left(2 p \right)}}{2} + \frac{1}{2}\right) - \frac{\cos{\left(2 p \right)}}{2} + \frac{1}{2}$$
2
1 - cos(2*p) / 2/p\\ 8/p\ 2/p\
------------ - 4*|1 - tan |-|| *cos |-|*tan |-|
2 \ \2// \2/ \2/
$$- 4 \left(- \tan^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2} \cos^{8}{\left(\frac{p}{2} \right)} \tan^{2}{\left(\frac{p}{2} \right)} + \frac{- \cos{\left(2 p \right)} + 1}{2}$$
/ 0 for p mod pi = 0 | | 4/p\ | 16*cot |-| | \2/ <-------------- otherwise | 4 |/ 2/p\\ ||1 + cot |-|| |\ \2// \
$$\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\frac{16 \cot^{4}{\left(\frac{p}{2} \right)}}{\left(\cot^{2}{\left(\frac{p}{2} \right)} + 1\right)^{4}} & \text{otherwise} \end{cases}$$
2 1 cos(2*p) / 2/p\\ 6/p\ 2/p\ - - -------- - 4*|1 - tan |-|| *cos |-|*sin |-| 2 2 \ \2// \2/ \2/
$$- 4 \left(- \tan^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2} \sin^{2}{\left(\frac{p}{2} \right)} \cos^{6}{\left(\frac{p}{2} \right)} - \frac{\cos{\left(2 p \right)}}{2} + \frac{1}{2}$$
2 2
1 - cos(2*p) / /p\\ / /p\\ 6/p\ 2/p\
------------ - 4*|1 + tan|-|| *|-1 + tan|-|| *cos |-|*sin |-|
2 \ \2// \ \2// \2/ \2/
$$- 4 \left(\tan{\left(\frac{p}{2} \right)} - 1\right)^{2} \left(\tan{\left(\frac{p}{2} \right)} + 1\right)^{2} \sin^{2}{\left(\frac{p}{2} \right)} \cos^{6}{\left(\frac{p}{2} \right)} + \frac{- \cos{\left(2 p \right)} + 1}{2}$$
2 8 1 cos(2*p) / 2/p\\ / 2/p\\ 16/p\ 2/p\ - - -------- - 4*|1 - tan |-|| *|1 - tan |-|| *cos |-|*tan |-| 2 2 \ \2// \ \4// \4/ \2/
$$- 4 \left(- \tan^{2}{\left(\frac{p}{4} \right)} + 1\right)^{8} \left(- \tan^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2} \cos^{16}{\left(\frac{p}{4} \right)} \tan^{2}{\left(\frac{p}{2} \right)} - \frac{\cos{\left(2 p \right)}}{2} + \frac{1}{2}$$
2
/ 2/p pi\\
| cos |- - --||
1 cos(2*p) | \2 2 /| 6/p\ 2/p pi\
- - -------- - 4*|1 - ------------| *cos |-|*cos |- - --|
2 2 | 2/p\ | \2/ \2 2 /
| cos |-| |
\ \2/ /
$$- 4 \left(1 - \frac{\cos^{2}{\left(\frac{p}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{p}{2} \right)}}\right)^{2} \cos^{6}{\left(\frac{p}{2} \right)} \cos^{2}{\left(\frac{p}{2} - \frac{\pi}{2} \right)} - \frac{\cos{\left(2 p \right)}}{2} + \frac{1}{2}$$
2
/ 2/p\ \
| sec |-| |
| \2/ |
4*|1 - ------------|
| 2/p pi\|
| sec |- - --||
1 1 \ \2 2 //
- - ---------- - ---------------------
2 2*sec(2*p) 6/p\ 2/p pi\
sec |-|*sec |- - --|
\2/ \2 2 /
$$\frac{1}{2} - \frac{1}{2 \sec{\left(2 p \right)}} - \frac{4 \left(- \frac{\sec^{2}{\left(\frac{p}{2} \right)}}{\sec^{2}{\left(\frac{p}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}}{\sec^{6}{\left(\frac{p}{2} \right)} \sec^{2}{\left(\frac{p}{2} - \frac{\pi}{2} \right)}}$$
2
2/p\ / 2/p\\ 2/p\
4*tan |-| 4*|1 - tan |-|| *tan |-|
\2/ \ \2// \2/
-------------- - ------------------------
2 4
/ 2/p\\ / 2/p\\
|1 + tan |-|| |1 + tan |-||
\ \2// \ \2//
$$- \frac{4 \left(- \tan^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2} \tan^{2}{\left(\frac{p}{2} \right)}}{\left(\tan^{2}{\left(\frac{p}{2} \right)} + 1\right)^{4}} + \frac{4 \tan^{2}{\left(\frac{p}{2} \right)}}{\left(\tan^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2}}$$
2
/ 4/p\\
| 4*sin |-||
| \2/| 4/p\ 8/pi p\
/pi \ 16*|1 - ---------| *sin |-|*sin |-- + -|
sin|-- + 2*p| | 2 | \2/ \2 2/
1 \2 / \ sin (p) /
- - ------------- - ----------------------------------------
2 2 2
sin (p)
$$- \frac{16 \left(- \frac{4 \sin^{4}{\left(\frac{p}{2} \right)}}{\sin^{2}{\left(p \right)}} + 1\right)^{2} \sin^{4}{\left(\frac{p}{2} \right)} \sin^{8}{\left(\frac{p}{2} + \frac{\pi}{2} \right)}}{\sin^{2}{\left(p \right)}} - \frac{\sin{\left(2 p + \frac{\pi}{2} \right)}}{2} + \frac{1}{2}$$
2
/ 2/pi p\\
| csc |-- - -||
| \2 2/|
4*|1 - ------------|
| 2/p\ |
| csc |-| |
1 1 \ \2/ /
- - --------------- - ---------------------
2 /pi \ 2/p\ 6/pi p\
2*csc|-- - 2*p| csc |-|*csc |-- - -|
\2 / \2/ \2 2/
$$\frac{1}{2} - \frac{1}{2 \csc{\left(- 2 p + \frac{\pi}{2} \right)}} - \frac{4 \left(1 - \frac{\csc^{2}{\left(- \frac{p}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{p}{2} \right)}}\right)^{2}}{\csc^{2}{\left(\frac{p}{2} \right)} \csc^{6}{\left(- \frac{p}{2} + \frac{\pi}{2} \right)}}$$
2 2 / 2 2 \ / 2 2 \ 1 sin (p) cos (p) |1 cos (p) sin (p)| |1 sin (p) cos (p)| - + ------- - ------- - |- + ------- - -------|*|- + ------- - -------| 2 2 2 \2 2 2 / \2 2 2 /
$$- \left(- \frac{\sin^{2}{\left(p \right)}}{2} + \frac{\cos^{2}{\left(p \right)}}{2} + \frac{1}{2}\right) \left(\frac{\sin^{2}{\left(p \right)}}{2} - \frac{\cos^{2}{\left(p \right)}}{2} + \frac{1}{2}\right) + \frac{\sin^{2}{\left(p \right)}}{2} - \frac{\cos^{2}{\left(p \right)}}{2} + \frac{1}{2}$$
2
/ 1 \
4*|1 - -------|
| 2/p\|
| cot |-||
4 \ \2//
---------------------- - ----------------------
2 4
/ 1 \ 2/p\ / 1 \ 2/p\
|1 + -------| *cot |-| |1 + -------| *cot |-|
| 2/p\| \2/ | 2/p\| \2/
| cot |-|| | cot |-||
\ \2// \ \2//
$$- \frac{4 \left(1 - \frac{1}{\cot^{2}{\left(\frac{p}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{p}{2} \right)}}\right)^{4} \cot^{2}{\left(\frac{p}{2} \right)}} + \frac{4}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{p}{2} \right)}}\right)^{2} \cot^{2}{\left(\frac{p}{2} \right)}}$$
2 8
/ 2/p\\ / 2/p\\ 2/p\
2 4*|1 - tan |-|| *|1 - tan |-|| *tan |-|
1 1 - tan (p) \ \2// \ \4// \2/
- - --------------- - ---------------------------------------
2 / 2 \ 8
2*\1 + tan (p)/ / 2/p\\
|1 + tan |-||
\ \4//
$$- \frac{4 \left(- \tan^{2}{\left(\frac{p}{4} \right)} + 1\right)^{8} \left(- \tan^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2} \tan^{2}{\left(\frac{p}{2} \right)}}{\left(\tan^{2}{\left(\frac{p}{4} \right)} + 1\right)^{8}} - \frac{- \tan^{2}{\left(p \right)} + 1}{2 \left(\tan^{2}{\left(p \right)} + 1\right)} + \frac{1}{2}$$
2 2
/ 2/p\\ / 2/p pi\\ 4/p\
|-1 + cot |-|| *|-1 + tan |- + --|| *sin |-|
4/p\ 2/p\ \ \2// \ \2 4 // \2/
- 4*cos |-| + 4*cos |-| - --------------------------------------------
\2/ \2/ 2
/ 2/p pi\\
|1 + tan |- + --||
\ \2 4 //
$$- \frac{\left(\tan^{2}{\left(\frac{p}{2} + \frac{\pi}{4} \right)} - 1\right)^{2} \left(\cot^{2}{\left(\frac{p}{2} \right)} - 1\right)^{2} \sin^{4}{\left(\frac{p}{2} \right)}}{\left(\tan^{2}{\left(\frac{p}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} - 4 \cos^{4}{\left(\frac{p}{2} \right)} + 4 \cos^{2}{\left(\frac{p}{2} \right)}$$
2/p\ 2/p\ 2/p pi\
4*tan |-| 16*tan |-|*tan |- + --|
\2/ \2/ \2 4 /
-------------- - ----------------------------------
2 2 2
/ 2/p\\ / 2/p\\ / 2/p pi\\
|1 + tan |-|| |1 + tan |-|| *|1 + tan |- + --||
\ \2// \ \2// \ \2 4 //
$$\frac{4 \tan^{2}{\left(\frac{p}{2} \right)}}{\left(\tan^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2}} - \frac{16 \tan^{2}{\left(\frac{p}{2} \right)} \tan^{2}{\left(\frac{p}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2} \left(\tan^{2}{\left(\frac{p}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
2/p\ 2/p\ 2/p pi\
4*cot |-| 16*cot |-|*tan |- + --|
\2/ \2/ \2 4 /
-------------- - ----------------------------------
2 2 2
/ 2/p\\ / 2/p\\ / 2/p pi\\
|1 + cot |-|| |1 + cot |-|| *|1 + tan |- + --||
\ \2// \ \2// \ \2 4 //
$$\frac{4 \cot^{2}{\left(\frac{p}{2} \right)}}{\left(\cot^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2}} - \frac{16 \tan^{2}{\left(\frac{p}{2} + \frac{\pi}{4} \right)} \cot^{2}{\left(\frac{p}{2} \right)}}{\left(\tan^{2}{\left(\frac{p}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\cot^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2}}$$
2 2 2
/ 2/p pi\\ 2 / 2/p pi\\ / 2/p\\ 2 4/p\
|1 - cot |- + --|| *(1 + sin(p)) |1 - cot |- + --|| *|1 - tan |-|| *(1 + sin(p)) *cos |-|
\ \2 4 // \ \2 4 // \ \2// \2/
--------------------------------- - --------------------------------------------------------
4 4
$$- \frac{\left(- \tan^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2} \left(- \cot^{2}{\left(\frac{p}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\sin{\left(p \right)} + 1\right)^{2} \cos^{4}{\left(\frac{p}{2} \right)}}{4} + \frac{\left(- \cot^{2}{\left(\frac{p}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\sin{\left(p \right)} + 1\right)^{2}}{4}$$
// 0 for p mod pi = 0\ // 1 for p mod 2*pi = 0\ // 0 for p mod pi = 0\ || | || | || | - |< 2 |*|< 2 | + |< 2 | ||sin (p) otherwise | ||cos (p) otherwise | ||sin (p) otherwise | \\ / \\ / \\ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\sin^{2}{\left(p \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: p \bmod 2 \pi = 0 \\\cos^{2}{\left(p \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\sin^{2}{\left(p \right)} & \text{otherwise} \end{cases}\right)$$
// 0 for p mod pi = 0\ // 1 for p mod 2*pi = 0\ // 0 for p mod pi = 0\ || | || | || | - |< 2 |*|< 2/ pi\ | + |< 2 | ||sin (p) otherwise | ||sin |p + --| otherwise | ||sin (p) otherwise | \\ / \\ \ 2 / / \\ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\sin^{2}{\left(p \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: p \bmod 2 \pi = 0 \\\sin^{2}{\left(p + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\sin^{2}{\left(p \right)} & \text{otherwise} \end{cases}\right)$$
// 0 for p mod pi = 0\ // 1 for p mod 2*pi = 0\ // 0 for p mod pi = 0\ || | || | || | - |< 2/ pi\ |*|< 2 | + |< 2/ pi\ | ||cos |p - --| otherwise | ||cos (p) otherwise | ||cos |p - --| otherwise | \\ \ 2 / / \\ / \\ \ 2 / /
$$\left(- \left(\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\cos^{2}{\left(p - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: p \bmod 2 \pi = 0 \\\cos^{2}{\left(p \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\cos^{2}{\left(p - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)$$
// 0 for p mod pi = 0\ // 1 for p mod 2*pi = 0\ // 0 for p mod pi = 0\ || | || | || | || 1 | || 1 | || 1 | - |<------- otherwise |*|<------------ otherwise | + |<------- otherwise | || 2 | || 2/pi \ | || 2 | ||csc (p) | ||csc |-- - p| | ||csc (p) | \\ / \\ \2 / / \\ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\frac{1}{\csc^{2}{\left(p \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: p \bmod 2 \pi = 0 \\\frac{1}{\csc^{2}{\left(- p + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\frac{1}{\csc^{2}{\left(p \right)}} & \text{otherwise} \end{cases}\right)$$
2 2 2
/ 2/p pi\\ / 2/p\\ / 2/p pi\\
|-1 + tan |- + --|| |-1 + cot |-|| *|-1 + tan |- + --||
\ \2 4 // \ \2// \ \2 4 //
-------------------- - ------------------------------------
2 2 2
/ 2/p pi\\ / 2/p\\ / 2/p pi\\
|1 + tan |- + --|| |1 + cot |-|| *|1 + tan |- + --||
\ \2 4 // \ \2// \ \2 4 //
$$- \frac{\left(\tan^{2}{\left(\frac{p}{2} + \frac{\pi}{4} \right)} - 1\right)^{2} \left(\cot^{2}{\left(\frac{p}{2} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{p}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\cot^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2}} + \frac{\left(\tan^{2}{\left(\frac{p}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{p}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
2
/ 4/p\\
| 4*sin |-||
| \2/| 4/p\
4/p\ 16*|1 - ---------| *sin |-|
16*sin |-| | 2 | \2/
\2/ \ sin (p) /
------------------------ - ---------------------------
2 4
/ 4/p\\ / 4/p\\
| 4*sin |-|| | 4*sin |-||
| \2/| 2 | \2/| 2
|1 + ---------| *sin (p) |1 + ---------| *sin (p)
| 2 | | 2 |
\ sin (p) / \ sin (p) /
$$- \frac{16 \left(- \frac{4 \sin^{4}{\left(\frac{p}{2} \right)}}{\sin^{2}{\left(p \right)}} + 1\right)^{2} \sin^{4}{\left(\frac{p}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{p}{2} \right)}}{\sin^{2}{\left(p \right)}} + 1\right)^{4} \sin^{2}{\left(p \right)}} + \frac{16 \sin^{4}{\left(\frac{p}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{p}{2} \right)}}{\sin^{2}{\left(p \right)}} + 1\right)^{2} \sin^{2}{\left(p \right)}}$$
2 2 2
/ 2/p pi\\ / 2/p pi\\ / 2/p\\
|1 - cot |- + --|| |1 - cot |- + --|| *|1 - tan |-||
\ \2 4 // \ \2 4 // \ \2//
------------------- - ----------------------------------
2 2 2
/ 2/p pi\\ / 2/p pi\\ / 2/p\\
|1 + cot |- + --|| |1 + cot |- + --|| *|1 + tan |-||
\ \2 4 // \ \2 4 // \ \2//
$$- \frac{\left(- \tan^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2} \left(- \cot^{2}{\left(\frac{p}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2} \left(\cot^{2}{\left(\frac{p}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} + \frac{\left(- \cot^{2}{\left(\frac{p}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}{\left(\cot^{2}{\left(\frac{p}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
// 0 for p mod pi = 0\ // 1 for p mod 2*pi = 0\ // 0 for p mod pi = 0\ || | || | || | || 1 | || 1 | || 1 | - |<------------ otherwise |*|<------- otherwise | + |<------------ otherwise | || 2/ pi\ | || 2 | || 2/ pi\ | ||sec |p - --| | ||sec (p) | ||sec |p - --| | \\ \ 2 / / \\ / \\ \ 2 / /
$$\left(- \left(\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\frac{1}{\sec^{2}{\left(p - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: p \bmod 2 \pi = 0 \\\frac{1}{\sec^{2}{\left(p \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\frac{1}{\sec^{2}{\left(p - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// p \
/ 1 for p mod pi = 0 || 1 for - mod 2*pi = 0|
< 2 || 2 |
1 \cos(2*p) otherwise / 2/p\\ 2/p\ || |
- - --------------------------- - 4*|1 - tan |-|| *tan |-|*|< 8 |
2 2 \ \2// \2/ ||/ 2/p\\ 16/p\ |
|||-1 + cot |-|| *sin |-| otherwise |
||\ \4// \4/ |
\\ /
$$\left(- 4 \left(- \tan^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2} \left(\begin{cases} 1 & \text{for}\: \frac{p}{2} \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{p}{4} \right)} - 1\right)^{8} \sin^{16}{\left(\frac{p}{4} \right)} & \text{otherwise} \end{cases}\right) \tan^{2}{\left(\frac{p}{2} \right)}\right) - \left(\frac{\begin{cases} 1 & \text{for}\: p \bmod \pi = 0 \\\cos{\left(2 p \right)} & \text{otherwise} \end{cases}}{2}\right) + \frac{1}{2}$$
// / pi\ \
// 0 for p mod pi = 0\ || 0 for |p + --| mod pi = 0| // 0 for p mod pi = 0\
|| | || \ 2 / | || |
- |< 2 |*|< | + |< 2 |
||sin (p) otherwise | || 2 2/p pi\ | ||sin (p) otherwise |
\\ / ||(1 + sin(p)) *cot |- + --| otherwise | \\ /
\\ \2 4 / /
$$\left(- \left(\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\sin^{2}{\left(p \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(p + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(p \right)} + 1\right)^{2} \cot^{2}{\left(\frac{p}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\sin^{2}{\left(p \right)} & \text{otherwise} \end{cases}\right)$$
// p \
|| 1 for - mod 2*pi = 0|
|| 2 |
|| |
2 || 8 |
/ 1 \ ||/ 2/p\\ |
/ 1 for p mod pi = 0 4*|1 - -------| *|<|-1 + cot |-|| |
| | 2/p\| ||\ \4// |
| 2 | cot |-|| ||--------------- otherwise |
<-1 + cot (p) \ \2// || 8 |
|------------ otherwise || / 2/p\\ |
| 2 || |1 + cot |-|| |
1 \1 + cot (p) \\ \ \4// /
- - ------------------------------- - -------------------------------------------------------
2 2 2/p\
cot |-|
\2/
$$\left(- \frac{4 \left(1 - \frac{1}{\cot^{2}{\left(\frac{p}{2} \right)}}\right)^{2} \left(\begin{cases} 1 & \text{for}\: \frac{p}{2} \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{p}{4} \right)} - 1\right)^{8}}{\left(\cot^{2}{\left(\frac{p}{4} \right)} + 1\right)^{8}} & \text{otherwise} \end{cases}\right)}{\cot^{2}{\left(\frac{p}{2} \right)}}\right) - \left(\frac{\begin{cases} 1 & \text{for}\: p \bmod \pi = 0 \\\frac{\cot^{2}{\left(p \right)} - 1}{\cot^{2}{\left(p \right)} + 1} & \text{otherwise} \end{cases}}{2}\right) + \frac{1}{2}$$
// / 3*pi\ \ // / 3*pi\ \
// 1 for p mod 2*pi = 0\ || 1 for |p + ----| mod 2*pi = 0| || 1 for |p + ----| mod 2*pi = 0|
|| | || \ 2 / | || \ 2 / |
- |< 2 |*|< | + |< |
||cos (p) otherwise | || 4/p\ 2/p\ | || 4/p\ 2/p\ |
\\ / ||- 4*cos |-| + 4*cos |-| otherwise | ||- 4*cos |-| + 4*cos |-| otherwise |
\\ \2/ \2/ / \\ \2/ \2/ /
$$\left(- \left(\begin{cases} 1 & \text{for}\: p \bmod 2 \pi = 0 \\\cos^{2}{\left(p \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(p + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\- 4 \cos^{4}{\left(\frac{p}{2} \right)} + 4 \cos^{2}{\left(\frac{p}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \left(p + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\- 4 \cos^{4}{\left(\frac{p}{2} \right)} + 4 \cos^{2}{\left(\frac{p}{2} \right)} & \text{otherwise} \end{cases}\right)$$
2
/ 2/p pi\\
| cos |- - --||
| \2 2 /| 2/p pi\
4*|1 - ------------| *cos |- - --|
2/p pi\ | 2/p\ | \2 2 /
4*cos |- - --| | cos |-| |
\2 2 / \ \2/ /
--------------------------- - ----------------------------------
2 4
/ 2/p pi\\ / 2/p pi\\
| cos |- - --|| | cos |- - --||
| \2 2 /| 2/p\ | \2 2 /| 2/p\
|1 + ------------| *cos |-| |1 + ------------| *cos |-|
| 2/p\ | \2/ | 2/p\ | \2/
| cos |-| | | cos |-| |
\ \2/ / \ \2/ /
$$- \frac{4 \left(1 - \frac{\cos^{2}{\left(\frac{p}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{p}{2} \right)}}\right)^{2} \cos^{2}{\left(\frac{p}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{p}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{p}{2} \right)}}\right)^{4} \cos^{2}{\left(\frac{p}{2} \right)}} + \frac{4 \cos^{2}{\left(\frac{p}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{p}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{p}{2} \right)}}\right)^{2} \cos^{2}{\left(\frac{p}{2} \right)}}$$
2
/ 2/p\ \
| sec |-| |
| \2/ | 2/p\
4*|1 - ------------| *sec |-|
2/p\ | 2/p pi\| \2/
4*sec |-| | sec |- - --||
\2/ \ \2 2 //
-------------------------------- - --------------------------------
2 4
/ 2/p\ \ / 2/p\ \
| sec |-| | | sec |-| |
| \2/ | 2/p pi\ | \2/ | 2/p pi\
|1 + ------------| *sec |- - --| |1 + ------------| *sec |- - --|
| 2/p pi\| \2 2 / | 2/p pi\| \2 2 /
| sec |- - --|| | sec |- - --||
\ \2 2 // \ \2 2 //
$$- \frac{4 \left(- \frac{\sec^{2}{\left(\frac{p}{2} \right)}}{\sec^{2}{\left(\frac{p}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec^{2}{\left(\frac{p}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{p}{2} \right)}}{\sec^{2}{\left(\frac{p}{2} - \frac{\pi}{2} \right)}} + 1\right)^{4} \sec^{2}{\left(\frac{p}{2} - \frac{\pi}{2} \right)}} + \frac{4 \sec^{2}{\left(\frac{p}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{p}{2} \right)}}{\sec^{2}{\left(\frac{p}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec^{2}{\left(\frac{p}{2} - \frac{\pi}{2} \right)}}$$
2
/ 2/pi p\\
| csc |-- - -||
| \2 2/| 2/pi p\
4*|1 - ------------| *csc |-- - -|
2/pi p\ | 2/p\ | \2 2/
4*csc |-- - -| | csc |-| |
\2 2/ \ \2/ /
--------------------------- - ----------------------------------
2 4
/ 2/pi p\\ / 2/pi p\\
| csc |-- - -|| | csc |-- - -||
| \2 2/| 2/p\ | \2 2/| 2/p\
|1 + ------------| *csc |-| |1 + ------------| *csc |-|
| 2/p\ | \2/ | 2/p\ | \2/
| csc |-| | | csc |-| |
\ \2/ / \ \2/ /
$$- \frac{4 \left(1 - \frac{\csc^{2}{\left(- \frac{p}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{p}{2} \right)}}\right)^{2} \csc^{2}{\left(- \frac{p}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{p}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{p}{2} \right)}}\right)^{4} \csc^{2}{\left(\frac{p}{2} \right)}} + \frac{4 \csc^{2}{\left(- \frac{p}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{p}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{p}{2} \right)}}\right)^{2} \csc^{2}{\left(\frac{p}{2} \right)}}$$
// 0 for p mod pi = 0\ // 1 for p mod 2*pi = 0\ // 0 for p mod pi = 0\ || | || | || | ||/ 0 for p mod pi = 0 | ||/ 1 for p mod 2*pi = 0 | ||/ 0 for p mod pi = 0 | - |<| |*|<| | + |<| | ||< 2 otherwise | ||< 2 otherwise | ||< 2 otherwise | |||sin (p) otherwise | |||cos (p) otherwise | |||sin (p) otherwise | \\\ / \\\ / \\\ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\sin^{2}{\left(p \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: p \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: p \bmod 2 \pi = 0 \\\cos^{2}{\left(p \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\sin^{2}{\left(p \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
// 0 for p mod pi = 0\ // 1 for p mod 2*pi = 0\ // 0 for p mod pi = 0\ || | || | || | || 2/p\ | || 2 | || 2/p\ | || 4*cot |-| | ||/ 2/p\\ | || 4*cot |-| | || \2/ | |||-1 + cot |-|| | || \2/ | - |<-------------- otherwise |*|<\ \2// | + |<-------------- otherwise | || 2 | ||--------------- otherwise | || 2 | ||/ 2/p\\ | || 2 | ||/ 2/p\\ | |||1 + cot |-|| | || / 2/p\\ | |||1 + cot |-|| | ||\ \2// | || |1 + cot |-|| | ||\ \2// | \\ / \\ \ \2// / \\ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{p}{2} \right)}}{\left(\cot^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: p \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{p}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{p}{2} \right)}}{\left(\cot^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// 0 for p mod pi = 0\ // 1 for p mod 2*pi = 0\ // 0 for p mod pi = 0\ || | || | || | || 2/p\ | || 2 | || 2/p\ | || 4*tan |-| | ||/ 2/p\\ | || 4*tan |-| | || \2/ | |||1 - tan |-|| | || \2/ | - |<-------------- otherwise |*|<\ \2// | + |<-------------- otherwise | || 2 | ||-------------- otherwise | || 2 | ||/ 2/p\\ | || 2 | ||/ 2/p\\ | |||1 + tan |-|| | ||/ 2/p\\ | |||1 + tan |-|| | ||\ \2// | |||1 + tan |-|| | ||\ \2// | \\ / \\\ \2// / \\ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\frac{4 \tan^{2}{\left(\frac{p}{2} \right)}}{\left(\tan^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: p \bmod 2 \pi = 0 \\\frac{\left(- \tan^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\frac{4 \tan^{2}{\left(\frac{p}{2} \right)}}{\left(\tan^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// / pi\ \
// 0 for p mod pi = 0\ || 0 for |p + --| mod pi = 0| // 0 for p mod pi = 0\
|| | || \ 2 / | || |
|| 2/p\ | || | || 2/p\ |
|| 4*cot |-| | || 2/p pi\ | || 4*cot |-| |
|| \2/ | || 4*cot |- + --| | || \2/ |
- |<-------------- otherwise |*|< \2 4 / | + |<-------------- otherwise |
|| 2 | ||------------------- otherwise | || 2 |
||/ 2/p\\ | || 2 | ||/ 2/p\\ |
|||1 + cot |-|| | ||/ 2/p pi\\ | |||1 + cot |-|| |
||\ \2// | |||1 + cot |- + --|| | ||\ \2// |
\\ / ||\ \2 4 // | \\ /
\\ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{p}{2} \right)}}{\left(\cot^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(p + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{p}{2} + \frac{\pi}{4} \right)}}{\left(\cot^{2}{\left(\frac{p}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{p}{2} \right)}}{\left(\cot^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// 1 for p mod 2*pi = 0\
|| |
// 0 for p mod pi = 0\ || 2 | // 0 for p mod pi = 0\
|| | ||/ 1 \ | || |
|| 4 | |||-1 + -------| | || 4 |
||---------------------- otherwise | ||| 2/p\| | ||---------------------- otherwise |
|| 2 | ||| tan |-|| | || 2 |
- |
$$\left(- \left(\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\frac{4}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{p}{2} \right)}}\right)^{2} \tan^{2}{\left(\frac{p}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: p \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{p}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{p}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\frac{4}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{p}{2} \right)}}\right)^{2} \tan^{2}{\left(\frac{p}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// / 3*pi\ \ // / 3*pi\ \
// 1 for p mod 2*pi = 0\ || 1 for |p + ----| mod 2*pi = 0| || 1 for |p + ----| mod 2*pi = 0|
|| | || \ 2 / | || \ 2 / |
|| 2 | || | || |
||/ 2/p\\ | || 2 | || 2 |
|||-1 + cot |-|| | ||/ 2/p pi\\ | ||/ 2/p pi\\ |
- |<\ \2// |*|<|-1 + tan |- + --|| | + |<|-1 + tan |- + --|| |
||--------------- otherwise | ||\ \2 4 // | ||\ \2 4 // |
|| 2 | ||-------------------- otherwise | ||-------------------- otherwise |
|| / 2/p\\ | || 2 | || 2 |
|| |1 + cot |-|| | ||/ 2/p pi\\ | ||/ 2/p pi\\ |
\\ \ \2// / |||1 + tan |- + --|| | |||1 + tan |- + --|| |
\\\ \2 4 // / \\\ \2 4 // /
$$\left(- \left(\begin{cases} 1 & \text{for}\: p \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{p}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(p + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\left(\tan^{2}{\left(\frac{p}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{p}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \left(p + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\left(\tan^{2}{\left(\frac{p}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{p}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// 0 for p mod pi = 0\ // 0 for p mod pi = 0\
|| | // 1 for p mod 2*pi = 0\ || |
|| 2 | || | || 2 |
|| sin (p) | || 2 | || sin (p) |
||------------------------ otherwise | ||/ 2 4/p\\ | ||------------------------ otherwise |
|| 2 | |||sin (p) - 4*sin |-|| | || 2 |
- |
$$\left(- \left(\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\frac{\sin^{2}{\left(p \right)}}{\left(1 + \frac{\sin^{2}{\left(p \right)}}{4 \sin^{4}{\left(\frac{p}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{p}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: p \bmod 2 \pi = 0 \\\frac{\left(- 4 \sin^{4}{\left(\frac{p}{2} \right)} + \sin^{2}{\left(p \right)}\right)^{2}}{\left(4 \sin^{4}{\left(\frac{p}{2} \right)} + \sin^{2}{\left(p \right)}\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\frac{\sin^{2}{\left(p \right)}}{\left(1 + \frac{\sin^{2}{\left(p \right)}}{4 \sin^{4}{\left(\frac{p}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{p}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// 1 for p mod 2*pi = 0\
|| |
// 0 for p mod pi = 0\ || 2 | // 0 for p mod pi = 0\
|| | ||/ 2 \ | || |
|| 2 | ||| sin (p) | | || 2 |
|| sin (p) | |||-1 + ---------| | || sin (p) |
||------------------------ otherwise | ||| 4/p\| | ||------------------------ otherwise |
|| 2 | ||| 4*sin |-|| | || 2 |
- |
$$\left(- \left(\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\frac{\sin^{2}{\left(p \right)}}{\left(1 + \frac{\sin^{2}{\left(p \right)}}{4 \sin^{4}{\left(\frac{p}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{p}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: p \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sin^{2}{\left(p \right)}}{4 \sin^{4}{\left(\frac{p}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sin^{2}{\left(p \right)}}{4 \sin^{4}{\left(\frac{p}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\frac{\sin^{2}{\left(p \right)}}{\left(1 + \frac{\sin^{2}{\left(p \right)}}{4 \sin^{4}{\left(\frac{p}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{p}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// 0 for p mod pi = 0\ // 1 for p mod 2*pi = 0\ // 0 for p mod pi = 0\ || | || | || | ||/ 0 for p mod pi = 0 | ||/ 1 for p mod 2*pi = 0 | ||/ 0 for p mod pi = 0 | ||| | ||| | ||| | ||| 2/p\ | ||| 2 | ||| 2/p\ | ||| 4*cot |-| | |||/ 2/p\\ | ||| 4*cot |-| | - |<| \2/ |*|<||-1 + cot |-|| | + |<| \2/ | ||<-------------- otherwise otherwise | ||<\ \2// otherwise | ||<-------------- otherwise otherwise | ||| 2 | |||--------------- otherwise | ||| 2 | |||/ 2/p\\ | ||| 2 | |||/ 2/p\\ | ||||1 + cot |-|| | ||| / 2/p\\ | ||||1 + cot |-|| | |||\ \2// | ||| |1 + cot |-|| | |||\ \2// | \\\ / \\\ \ \2// / \\\ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{p}{2} \right)}}{\left(\cot^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: p \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: p \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{p}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{p}{2} \right)}}{\left(\cot^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
// 1 for p mod 2*pi = 0\
|| |
// 0 for p mod pi = 0\ || 2 | // 0 for p mod pi = 0\
|| | ||/ 2/p\ \ | || |
|| 2/p\ | ||| cos |-| | | || 2/p\ |
|| 4*cos |-| | ||| \2/ | | || 4*cos |-| |
|| \2/ | |||-1 + ------------| | || \2/ |
||-------------------------------- otherwise | ||| 2/p pi\| | ||-------------------------------- otherwise |
|| 2 | ||| cos |- - --|| | || 2 |
- |
$$\left(- \left(\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\frac{4 \cos^{2}{\left(\frac{p}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{p}{2} \right)}}{\cos^{2}{\left(\frac{p}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \cos^{2}{\left(\frac{p}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: p \bmod 2 \pi = 0 \\\frac{\left(\frac{\cos^{2}{\left(\frac{p}{2} \right)}}{\cos^{2}{\left(\frac{p}{2} - \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\cos^{2}{\left(\frac{p}{2} \right)}}{\cos^{2}{\left(\frac{p}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\frac{4 \cos^{2}{\left(\frac{p}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{p}{2} \right)}}{\cos^{2}{\left(\frac{p}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \cos^{2}{\left(\frac{p}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// 1 for p mod 2*pi = 0\
|| |
// 0 for p mod pi = 0\ || 2 | // 0 for p mod pi = 0\
|| | ||/ 2/p pi\\ | || |
|| 2/p pi\ | ||| sec |- - --|| | || 2/p pi\ |
|| 4*sec |- - --| | ||| \2 2 /| | || 4*sec |- - --| |
|| \2 2 / | |||-1 + ------------| | || \2 2 / |
||--------------------------- otherwise | ||| 2/p\ | | ||--------------------------- otherwise |
|| 2 | ||| sec |-| | | || 2 |
- |
$$\left(- \left(\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\frac{4 \sec^{2}{\left(\frac{p}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{p}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{p}{2} \right)}}\right)^{2} \sec^{2}{\left(\frac{p}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: p \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{p}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{p}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sec^{2}{\left(\frac{p}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{p}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\frac{4 \sec^{2}{\left(\frac{p}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{p}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{p}{2} \right)}}\right)^{2} \sec^{2}{\left(\frac{p}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// 1 for p mod 2*pi = 0\
|| |
// 0 for p mod pi = 0\ || 2 | // 0 for p mod pi = 0\
|| | ||/ 2/p\ \ | || |
|| 2/p\ | ||| csc |-| | | || 2/p\ |
|| 4*csc |-| | ||| \2/ | | || 4*csc |-| |
|| \2/ | |||-1 + ------------| | || \2/ |
||-------------------------------- otherwise | ||| 2/pi p\| | ||-------------------------------- otherwise |
|| 2 | ||| csc |-- - -|| | || 2 |
- |
$$\left(- \left(\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\frac{4 \csc^{2}{\left(\frac{p}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{p}{2} \right)}}{\csc^{2}{\left(- \frac{p}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2} \csc^{2}{\left(- \frac{p}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: p \bmod 2 \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{p}{2} \right)}}{\csc^{2}{\left(- \frac{p}{2} + \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\csc^{2}{\left(\frac{p}{2} \right)}}{\csc^{2}{\left(- \frac{p}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\frac{4 \csc^{2}{\left(\frac{p}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{p}{2} \right)}}{\csc^{2}{\left(- \frac{p}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2} \csc^{2}{\left(- \frac{p}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
-Piecewise((0, Mod(p = pi, 0)), (4*csc(p/2)^2/((1 + csc(p/2)^2/csc(pi/2 - p/2)^2)^2*csc(pi/2 - p/2)^2), True))*Piecewise((1, Mod(p = 2*pi, 0)), ((-1 + csc(p/2)^2/csc(pi/2 - p/2)^2)^2/(1 + csc(p/2)^2/csc(pi/2 - p/2)^2)^2, True)) + Piecewise((0, Mod(p = pi, 0)), (4*csc(p/2)^2/((1 + csc(p/2)^2/csc(pi/2 - p/2)^2)^2*csc(pi/2 - p/2)^2), True))
Степени
[src]
2
/ I*p -I*p\ 2
2 |e e | / -I*p I*p\
/ -I*p I*p\ |---- + -----| *\- e + e /
\- e + e / \ 2 2 /
- ----------------- + ---------------------------------
4 4
$$\frac{\left(\frac{e^{i p}}{2} + \frac{e^{- i p}}{2}\right)^{2} \left(e^{i p} - e^{- i p}\right)^{2}}{4} - \frac{\left(e^{i p} - e^{- i p}\right)^{2}}{4}$$
-(-exp(-i*p) + exp(i*p))^2/4 + (exp(i*p)/2 + exp(-i*p)/2)^2*(-exp(-i*p) + exp(i*p))^2/4
Объединение рациональных выражений
[src]
2 / 2 \ sin (p)*\1 - cos (p)/
$$\left(- \cos^{2}{\left(p \right)} + 1\right) \sin^{2}{\left(p \right)}$$
sin(p)^2*(1 - cos(p)^2)
Комбинаторика
[src]
2 -sin (p)*(1 + cos(p))*(-1 + cos(p))
$$- \left(\cos{\left(p \right)} - 1\right) \left(\cos{\left(p \right)} + 1\right) \sin^{2}{\left(p \right)}$$
-sin(p)^2*(1 + cos(p))*(-1 + cos(p))