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