Господин Экзамен
Общий знаменатель cos(p)/7*cos(p)/42-sin(p)/7*sin(p)/42
Решение
Вы ввели
[src]
cos(p)*cos(p) sin(p)*sin(p)
------------- - -------------
7*42 7*42
$$- \frac{\sin{\left(p \right)} \sin{\left(p \right)}}{7 \cdot 42} + \frac{\cos{\left(p \right)} \cos{\left(p \right)}}{7 \cdot 42}$$
cos(p)*cos(p)/(7*42) - sin(p)*sin(p)/(7*42)
Рациональный знаменатель
[src]
2 2
cos (p) - sin (p)
-----------------
294
$$\frac{- \sin^{2}{\left(p \right)} + \cos^{2}{\left(p \right)}}{294}$$
2 2
sin (p) cos (p)
- ------- + -------
294 294
$$- \frac{\sin^{2}{\left(p \right)}}{294} + \frac{\cos^{2}{\left(p \right)}}{294}$$
-sin(p)^2/294 + cos(p)^2/294
Комбинаторика
[src]
(-sin(p) + cos(p))*(cos(p) + sin(p))
------------------------------------
294
$$\frac{\left(- \sin{\left(p \right)} + \cos{\left(p \right)}\right) \left(\sin{\left(p \right)} + \cos{\left(p \right)}\right)}{294}$$
(-sin(p) + cos(p))*(cos(p) + sin(p))/294
Численный ответ
[src]
0.00340136054421769*cos(p)^2 - 0.00340136054421769*sin(p)^2
0.00340136054421769*cos(p)^2 - 0.00340136054421769*sin(p)^2
Объединение рациональных выражений
[src]
2 2
cos (p) - sin (p)
-----------------
294
$$\frac{- \sin^{2}{\left(p \right)} + \cos^{2}{\left(p \right)}}{294}$$
(cos(p)^2 - sin(p)^2)/294
Степени
[src]
2 2
sin (p) cos (p)
- ------- + -------
294 294
$$- \frac{\sin^{2}{\left(p \right)}}{294} + \frac{\cos^{2}{\left(p \right)}}{294}$$
2
/ I*p -I*p\
|e e | 2
|---- + -----| / -I*p I*p\
\ 2 2 / \- e + e /
--------------- + -----------------
294 1176
$$\frac{\left(\frac{e^{i p}}{2} + \frac{e^{- i p}}{2}\right)^{2}}{294} + \frac{\left(e^{i p} - e^{- i p}\right)^{2}}{1176}$$
(exp(i*p)/2 + exp(-i*p)/2)^2/294 + (-exp(-i*p) + exp(i*p))^2/1176
Общий знаменатель
[src]
2 2
sin (p) cos (p)
- ------- + -------
294 294
$$- \frac{\sin^{2}{\left(p \right)}}{294} + \frac{\cos^{2}{\left(p \right)}}{294}$$
-sin(p)^2/294 + cos(p)^2/294
Собрать выражение
[src]
cos(2*p) -------- 294
$$\frac{\cos{\left(2 p \right)}}{294}$$
2 2
sin (p) cos (p)
- ------- + -------
294 294
$$- \frac{\sin^{2}{\left(p \right)}}{294} + \frac{\cos^{2}{\left(p \right)}}{294}$$
-sin(p)^2/294 + cos(p)^2/294
Тригонометрическая часть
[src]
cos(2*p) -------- 294
$$\frac{\cos{\left(2 p \right)}}{294}$$
1 ------------ 294*sec(2*p)
$$\frac{1}{294 \sec{\left(2 p \right)}}$$
/pi \
sin|-- + 2*p|
\2 /
-------------
294
$$\frac{\sin{\left(2 p + \frac{\pi}{2} \right)}}{294}$$
1
-----------------
/pi \
294*csc|-- - 2*p|
\2 /
$$\frac{1}{294 \csc{\left(- 2 p + \frac{\pi}{2} \right)}}$$
2 2
sin (p) cos (p)
- ------- + -------
294 294
$$- \frac{\sin^{2}{\left(p \right)}}{294} + \frac{\cos^{2}{\left(p \right)}}{294}$$
1 1
- ----------- + -----------
2 2
294*csc (p) 294*sec (p)
$$\frac{1}{294 \sec^{2}{\left(p \right)}} - \frac{1}{294 \csc^{2}{\left(p \right)}}$$
2
1 - tan (p)
-----------------
/ 2 \
294*\1 + tan (p)/
$$\frac{- \tan^{2}{\left(p \right)} + 1}{294 \left(\tan^{2}{\left(p \right)} + 1\right)}$$
2/ pi\
cos |p - --| 2
\ 2 / cos (p)
- ------------ + -------
294 294
$$\frac{\cos^{2}{\left(p \right)}}{294} - \frac{\cos^{2}{\left(p - \frac{\pi}{2} \right)}}{294}$$
2/ pi\
2 sin |p + --|
sin (p) \ 2 /
- ------- + ------------
294 294
$$- \frac{\sin^{2}{\left(p \right)}}{294} + \frac{\sin^{2}{\left(p + \frac{\pi}{2} \right)}}{294}$$
1 1
- ----------- + ----------------
2 2/pi \
294*csc (p) 294*csc |-- - p|
\2 /
$$\frac{1}{294 \csc^{2}{\left(- p + \frac{\pi}{2} \right)}} - \frac{1}{294 \csc^{2}{\left(p \right)}}$$
1 1
- ---------------- + -----------
2/ pi\ 2
294*sec |p - --| 294*sec (p)
\ 2 /
$$- \frac{1}{294 \sec^{2}{\left(p - \frac{\pi}{2} \right)}} + \frac{1}{294 \sec^{2}{\left(p \right)}}$$
1 1
- ---------------- + -----------
2/pi \ 2
294*sec |-- - p| 294*sec (p)
\2 /
$$- \frac{1}{294 \sec^{2}{\left(- p + \frac{\pi}{2} \right)}} + \frac{1}{294 \sec^{2}{\left(p \right)}}$$
1 1
- ---------------- + ----------------
2 2/pi \
294*csc (pi - p) 294*csc |-- - p|
\2 /
$$\frac{1}{294 \csc^{2}{\left(- p + \frac{\pi}{2} \right)}} - \frac{1}{294 \csc^{2}{\left(- p + \pi \right)}}$$
/ 1 for p mod pi = 0
<
\cos(2*p) otherwise
---------------------------
294
$$\frac{\begin{cases} 1 & \text{for}\: p \bmod \pi = 0 \\\cos{\left(2 p \right)} & \text{otherwise} \end{cases}}{294}$$
2 1 cos(p) (1 - cos(p)) cos(2*p) - --- + ------ + ------------- + -------- 196 147 294 588
$$\frac{\left(- \cos{\left(p \right)} + 1\right)^{2}}{294} + \frac{\cos{\left(p \right)}}{147} + \frac{\cos{\left(2 p \right)}}{588} - \frac{1}{196}$$
/ 1 for p mod pi = 0
|
| 2
<-1 + cot (p)
|------------ otherwise
| 2
\1 + cot (p)
-------------------------------
294
$$\frac{\begin{cases} 1 & \text{for}\: p \bmod \pi = 0 \\\frac{\cot^{2}{\left(p \right)} - 1}{\cot^{2}{\left(p \right)} + 1} & \text{otherwise} \end{cases}}{294}$$
2
/ 2/p pi\\ 2
|1 - cot |- + --|| *(1 + sin(p))
1 + cos(2*p) \ \2 4 //
------------ - ---------------------------------
588 1176
$$- \frac{\left(- \cot^{2}{\left(\frac{p}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\sin{\left(p \right)} + 1\right)^{2}}{1176} + \frac{\cos{\left(2 p \right)} + 1}{588}$$
2/p pi\
2*tan |- + --|
1 - cos(2*p) \2 4 /
- ------------ + -----------------------
588 2
/ 2/p pi\\
147*|1 + tan |- + --||
\ \2 4 //
$$- \frac{- \cos{\left(2 p \right)} + 1}{588} + \frac{2 \tan^{2}{\left(\frac{p}{2} + \frac{\pi}{4} \right)}}{147 \left(\tan^{2}{\left(\frac{p}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
4 8/p\ 2 4/p\
sin (p) + 16*sin |-| - 24*sin (p)*sin |-|
\2/ \2/
-----------------------------------------
2
/ 2 4/p\\
294*|sin (p) + 4*sin |-||
\ \2//
$$\frac{16 \sin^{8}{\left(\frac{p}{2} \right)} - 24 \sin^{4}{\left(\frac{p}{2} \right)} \sin^{2}{\left(p \right)} + \sin^{4}{\left(p \right)}}{294 \left(4 \sin^{4}{\left(\frac{p}{2} \right)} + \sin^{2}{\left(p \right)}\right)^{2}}$$
2
2/p\ / 2/p\\
2*tan |-| |1 - tan |-||
\2/ \ \2//
- ------------------ + ------------------
2 2
/ 2/p\\ / 2/p\\
147*|1 + tan |-|| 294*|1 + tan |-||
\ \2// \ \2//
$$\frac{\left(- \tan^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2}}{294 \left(\tan^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2}} - \frac{2 \tan^{2}{\left(\frac{p}{2} \right)}}{147 \left(\tan^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2}}$$
2/p\ 2/p pi\
2*tan |-| 2*tan |- + --|
\2/ \2 4 /
- ------------------ + -----------------------
2 2
/ 2/p\\ / 2/p pi\\
147*|1 + tan |-|| 147*|1 + tan |- + --||
\ \2// \ \2 4 //
$$\frac{2 \tan^{2}{\left(\frac{p}{2} + \frac{\pi}{4} \right)}}{147 \left(\tan^{2}{\left(\frac{p}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} - \frac{2 \tan^{2}{\left(\frac{p}{2} \right)}}{147 \left(\tan^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2}}$$
2/p\ 2/p pi\
2*cot |-| 2*tan |- + --|
\2/ \2 4 /
- ------------------ + -----------------------
2 2
/ 2/p\\ / 2/p pi\\
147*|1 + cot |-|| 147*|1 + tan |- + --||
\ \2// \ \2 4 //
$$- \frac{2 \cot^{2}{\left(\frac{p}{2} \right)}}{147 \left(\cot^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2}} + \frac{2 \tan^{2}{\left(\frac{p}{2} + \frac{\pi}{4} \right)}}{147 \left(\tan^{2}{\left(\frac{p}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
2
/ 1 \
|1 - -------|
| 2/p\|
| cot |-||
2 \ \2//
- -------------------------- + ------------------
2 2
/ 1 \ 2/p\ / 1 \
147*|1 + -------| *cot |-| 294*|1 + -------|
| 2/p\| \2/ | 2/p\|
| cot |-|| | cot |-||
\ \2// \ \2//
$$\frac{\left(1 - \frac{1}{\cot^{2}{\left(\frac{p}{2} \right)}}\right)^{2}}{294 \left(1 + \frac{1}{\cot^{2}{\left(\frac{p}{2} \right)}}\right)^{2}} - \frac{2}{147 \left(1 + \frac{1}{\cot^{2}{\left(\frac{p}{2} \right)}}\right)^{2} \cot^{2}{\left(\frac{p}{2} \right)}}$$
/ 0 for p mod pi = 0 / 1 for p mod 2*pi = 0
| |
< 2 < 2
|sin (p) otherwise |cos (p) otherwise
\ \
- -------------------------- + ----------------------------
294 294
$$\left(- \frac{\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\sin^{2}{\left(p \right)} & \text{otherwise} \end{cases}}{294}\right) + \left(\frac{\begin{cases} 1 & \text{for}\: p \bmod 2 \pi = 0 \\\cos^{2}{\left(p \right)} & \text{otherwise} \end{cases}}{294}\right)$$
/ 0 for p mod pi = 0 / 1 for p mod 2*pi = 0
| |
< 2/ pi\ < 2
|cos |p - --| otherwise |cos (p) otherwise
\ \ 2 / \
- ------------------------------- + ----------------------------
294 294
$$\left(- \frac{\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\cos^{2}{\left(p - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}}{294}\right) + \left(\frac{\begin{cases} 1 & \text{for}\: p \bmod 2 \pi = 0 \\\cos^{2}{\left(p \right)} & \text{otherwise} \end{cases}}{294}\right)$$
/ 0 for p mod pi = 0 / 1 for p mod 2*pi = 0
| |
< 2 < 2/ pi\
|sin (p) otherwise |sin |p + --| otherwise
\ \ \ 2 /
- -------------------------- + ---------------------------------
294 294
$$\left(- \frac{\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\sin^{2}{\left(p \right)} & \text{otherwise} \end{cases}}{294}\right) + \left(\frac{\begin{cases} 1 & \text{for}\: p \bmod 2 \pi = 0 \\\sin^{2}{\left(p + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}}{294}\right)$$
2 2
/ 2/p pi\\ / 2/p\\
|-1 + tan |- + --|| |-1 + cot |-||
\ \2 4 // \ \2//
- ----------------------- + ------------------
2 2
/ 2/p pi\\ / 2/p\\
294*|1 + tan |- + --|| 294*|1 + cot |-||
\ \2 4 // \ \2//
$$- \frac{\left(\tan^{2}{\left(\frac{p}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{294 \left(\tan^{2}{\left(\frac{p}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} + \frac{\left(\cot^{2}{\left(\frac{p}{2} \right)} - 1\right)^{2}}{294 \left(\cot^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2}}$$
2 2
/ 2/p pi\\ / 2/p\\
|1 - cot |- + --|| |1 - tan |-||
\ \2 4 // \ \2//
- ----------------------- + ------------------
2 2
/ 2/p pi\\ / 2/p\\
294*|1 + cot |- + --|| 294*|1 + tan |-||
\ \2 4 // \ \2//
$$\frac{\left(- \tan^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2}}{294 \left(\tan^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2}} - \frac{\left(- \cot^{2}{\left(\frac{p}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}{294 \left(\cot^{2}{\left(\frac{p}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
/ 0 for p mod pi = 0 / 1 for p mod 2*pi = 0
| |
| 1 | 1
<------------ otherwise <------- otherwise
| 2/ pi\ | 2
|sec |p - --| |sec (p)
\ \ 2 / \
- ------------------------------- + ----------------------------
294 294
$$\left(- \frac{\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\frac{1}{\sec^{2}{\left(p - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}}{294}\right) + \left(\frac{\begin{cases} 1 & \text{for}\: p \bmod 2 \pi = 0 \\\frac{1}{\sec^{2}{\left(p \right)}} & \text{otherwise} \end{cases}}{294}\right)$$
/ 0 for p mod pi = 0 / 1 for p mod 2*pi = 0
| |
| 1 | 1
<------- otherwise <------------ otherwise
| 2 | 2/pi \
|csc (p) |csc |-- - p|
\ \ \2 /
- -------------------------- + ---------------------------------
294 294
$$\left(- \frac{\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\frac{1}{\csc^{2}{\left(p \right)}} & \text{otherwise} \end{cases}}{294}\right) + \left(\frac{\begin{cases} 1 & \text{for}\: p \bmod 2 \pi = 0 \\\frac{1}{\csc^{2}{\left(- p + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}}{294}\right)$$
/ / 3*pi\
| 1 for |p + ----| mod 2*pi = 0
| \ 2 / / 1 for p mod 2*pi = 0
< |
| 4/p\ 2/p\ < 2
|- 4*cos |-| + 4*cos |-| otherwise |cos (p) otherwise
\ \2/ \2/ \
- ----------------------------------------------------- + ----------------------------
294 294
$$\left(\frac{\begin{cases} 1 & \text{for}\: p \bmod 2 \pi = 0 \\\cos^{2}{\left(p \right)} & \text{otherwise} \end{cases}}{294}\right) - \left(\frac{\begin{cases} 1 & \text{for}\: \left(p + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\- 4 \cos^{4}{\left(\frac{p}{2} \right)} + 4 \cos^{2}{\left(\frac{p}{2} \right)} & \text{otherwise} \end{cases}}{294}\right)$$
/ / pi\
| 0 for |p + --| mod pi = 0
/ 0 for p mod pi = 0 | \ 2 /
| <
< 2 | 2 2/p pi\
|sin (p) otherwise |(1 + sin(p)) *cot |- + --| otherwise
\ \ \2 4 /
- -------------------------- + ----------------------------------------------------
294 294
$$\left(- \frac{\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\sin^{2}{\left(p \right)} & \text{otherwise} \end{cases}}{294}\right) + \left(\frac{\begin{cases} 0 & \text{for}\: \left(p + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(p \right)} + 1\right)^{2} \cot^{2}{\left(\frac{p}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}}{294}\right)$$
2
/ 4/p\\
| 4*sin |-||
| \2/|
|1 - ---------| 4/p\
| 2 | 8*sin |-|
\ sin (p) / \2/
-------------------- - ----------------------------
2 2
/ 4/p\\ / 4/p\\
| 4*sin |-|| | 4*sin |-||
| \2/| | \2/| 2
294*|1 + ---------| 147*|1 + ---------| *sin (p)
| 2 | | 2 |
\ sin (p) / \ sin (p) /
$$\frac{\left(- \frac{4 \sin^{4}{\left(\frac{p}{2} \right)}}{\sin^{2}{\left(p \right)}} + 1\right)^{2}}{294 \left(\frac{4 \sin^{4}{\left(\frac{p}{2} \right)}}{\sin^{2}{\left(p \right)}} + 1\right)^{2}} - \frac{8 \sin^{4}{\left(\frac{p}{2} \right)}}{147 \left(\frac{4 \sin^{4}{\left(\frac{p}{2} \right)}}{\sin^{2}{\left(p \right)}} + 1\right)^{2} \sin^{2}{\left(p \right)}}$$
/ 0 for p mod pi = 0 / 1 for p mod 2*pi = 0
| |
|/ 0 for p mod pi = 0 |/ 1 for p mod 2*pi = 0
<| <|
|< 2 otherwise |< 2 otherwise
||sin (p) otherwise ||cos (p) otherwise
\\ \\
- --------------------------------------------- + -------------------------------------------------
294 294
$$\left(- \frac{\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\sin^{2}{\left(p \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}}{294}\right) + \left(\frac{\begin{cases} 1 & \text{for}\: p \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: p \bmod 2 \pi = 0 \\\cos^{2}{\left(p \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}}{294}\right)$$
/ 0 for p mod pi = 0 / 1 for p mod 2*pi = 0
| |
| 2/p\ | 2
| 4*cot |-| |/ 2/p\\
| \2/ ||-1 + cot |-||
<-------------- otherwise <\ \2//
| 2 |--------------- otherwise
|/ 2/p\\ | 2
||1 + cot |-|| | / 2/p\\
|\ \2// | |1 + cot |-||
\ \ \ \2//
- --------------------------------- + ------------------------------------
294 294
$$\left(- \frac{\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{p}{2} \right)}}{\left(\cot^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}}{294}\right) + \left(\frac{\begin{cases} 1 & \text{for}\: p \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{p}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}}{294}\right)$$
/ 0 for p mod pi = 0 / 1 for p mod 2*pi = 0
| |
| 2/p\ | 2
| 4*tan |-| |/ 2/p\\
| \2/ ||1 - tan |-||
<-------------- otherwise <\ \2//
| 2 |-------------- otherwise
|/ 2/p\\ | 2
||1 + tan |-|| |/ 2/p\\
|\ \2// ||1 + tan |-||
\ \\ \2//
- --------------------------------- + -----------------------------------
294 294
$$\left(- \frac{\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\frac{4 \tan^{2}{\left(\frac{p}{2} \right)}}{\left(\tan^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}}{294}\right) + \left(\frac{\begin{cases} 1 & \text{for}\: p \bmod 2 \pi = 0 \\\frac{\left(- \tan^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}}{294}\right)$$
2
/ 2/p pi\\
| cos |- - --||
| \2 2 /|
|1 - ------------|
| 2/p\ | 2/p pi\
| cos |-| | 2*cos |- - --|
\ \2/ / \2 2 /
----------------------- - -------------------------------
2 2
/ 2/p pi\\ / 2/p pi\\
| cos |- - --|| | cos |- - --||
| \2 2 /| | \2 2 /| 2/p\
294*|1 + ------------| 147*|1 + ------------| *cos |-|
| 2/p\ | | 2/p\ | \2/
| cos |-| | | cos |-| |
\ \2/ / \ \2/ /
$$\frac{\left(1 - \frac{\cos^{2}{\left(\frac{p}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{p}{2} \right)}}\right)^{2}}{294 \left(1 + \frac{\cos^{2}{\left(\frac{p}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{p}{2} \right)}}\right)^{2}} - \frac{2 \cos^{2}{\left(\frac{p}{2} - \frac{\pi}{2} \right)}}{147 \left(1 + \frac{\cos^{2}{\left(\frac{p}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{p}{2} \right)}}\right)^{2} \cos^{2}{\left(\frac{p}{2} \right)}}$$
2
/ 2/p\ \
| sec |-| |
| \2/ |
|1 - ------------|
| 2/p pi\| 2/p\
| sec |- - --|| 2*sec |-|
\ \2 2 // \2/
----------------------- - ------------------------------------
2 2
/ 2/p\ \ / 2/p\ \
| sec |-| | | sec |-| |
| \2/ | | \2/ | 2/p pi\
294*|1 + ------------| 147*|1 + ------------| *sec |- - --|
| 2/p pi\| | 2/p pi\| \2 2 /
| sec |- - --|| | sec |- - --||
\ \2 2 // \ \2 2 //
$$\frac{\left(- \frac{\sec^{2}{\left(\frac{p}{2} \right)}}{\sec^{2}{\left(\frac{p}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}}{294 \left(\frac{\sec^{2}{\left(\frac{p}{2} \right)}}{\sec^{2}{\left(\frac{p}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}} - \frac{2 \sec^{2}{\left(\frac{p}{2} \right)}}{147 \left(\frac{\sec^{2}{\left(\frac{p}{2} \right)}}{\sec^{2}{\left(\frac{p}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec^{2}{\left(\frac{p}{2} - \frac{\pi}{2} \right)}}$$
/ 1 for p mod 2*pi = 0
|
| 2
|/ 1 \
/ 0 for p mod pi = 0 ||-1 + -------|
| || 2/p\|
| 4 || tan |-||
|---------------------- otherwise <\ \2//
| 2 |--------------- otherwise
$$\left(- \frac{\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\frac{4}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{p}{2} \right)}}\right)^{2} \tan^{2}{\left(\frac{p}{2} \right)}} & \text{otherwise} \end{cases}}{294}\right) + \left(\frac{\begin{cases} 1 & \text{for}\: p \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{p}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{p}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}}{294}\right)$$
2
/ 2/pi p\\
| csc |-- - -||
| \2 2/|
|1 - ------------|
| 2/p\ | 2/pi p\
| csc |-| | 2*csc |-- - -|
\ \2/ / \2 2/
----------------------- - -------------------------------
2 2
/ 2/pi p\\ / 2/pi p\\
| csc |-- - -|| | csc |-- - -||
| \2 2/| | \2 2/| 2/p\
294*|1 + ------------| 147*|1 + ------------| *csc |-|
| 2/p\ | | 2/p\ | \2/
| csc |-| | | csc |-| |
\ \2/ / \ \2/ /
$$\frac{\left(1 - \frac{\csc^{2}{\left(- \frac{p}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{p}{2} \right)}}\right)^{2}}{294 \left(1 + \frac{\csc^{2}{\left(- \frac{p}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{p}{2} \right)}}\right)^{2}} - \frac{2 \csc^{2}{\left(- \frac{p}{2} + \frac{\pi}{2} \right)}}{147 \left(1 + \frac{\csc^{2}{\left(- \frac{p}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{p}{2} \right)}}\right)^{2} \csc^{2}{\left(\frac{p}{2} \right)}}$$
/ / pi\
| 0 for |p + --| mod pi = 0
/ 0 for p mod pi = 0 | \ 2 /
| |
| 2/p\ | 2/p pi\
| 4*cot |-| | 4*cot |- + --|
| \2/ < \2 4 /
<-------------- otherwise |------------------- otherwise
| 2 | 2
|/ 2/p\\ |/ 2/p pi\\
||1 + cot |-|| ||1 + cot |- + --||
|\ \2// |\ \2 4 //
\ \
- --------------------------------- + ---------------------------------------------
294 294
$$\left(- \frac{\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{p}{2} \right)}}{\left(\cot^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}}{294}\right) + \left(\frac{\begin{cases} 0 & \text{for}\: \left(p + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{p}{2} + \frac{\pi}{4} \right)}}{\left(\cot^{2}{\left(\frac{p}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}}{294}\right)$$
/ / 3*pi\
| 1 for |p + ----| mod 2*pi = 0
| \ 2 / / 1 for p mod 2*pi = 0
| |
| 2 | 2
|/ 2/p pi\\ |/ 2/p\\
<|-1 + tan |- + --|| ||-1 + cot |-||
|\ \2 4 // <\ \2//
|-------------------- otherwise |--------------- otherwise
| 2 | 2
|/ 2/p pi\\ | / 2/p\\
||1 + tan |- + --|| | |1 + cot |-||
\\ \2 4 // \ \ \2//
- -------------------------------------------------- + ------------------------------------
294 294
$$\left(\frac{\begin{cases} 1 & \text{for}\: p \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{p}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}}{294}\right) - \left(\frac{\begin{cases} 1 & \text{for}\: \left(p + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\left(\tan^{2}{\left(\frac{p}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{p}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}}{294}\right)$$
/ 0 for p mod pi = 0
|
| 2 / 1 for p mod 2*pi = 0
| sin (p) |
|------------------------ otherwise | 2
| 2 |/ 2 4/p\\
$$\left(- \frac{\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\frac{\sin^{2}{\left(p \right)}}{\left(1 + \frac{\sin^{2}{\left(p \right)}}{4 \sin^{4}{\left(\frac{p}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{p}{2} \right)}} & \text{otherwise} \end{cases}}{294}\right) + \left(\frac{\begin{cases} 1 & \text{for}\: p \bmod 2 \pi = 0 \\\frac{\left(- 4 \sin^{4}{\left(\frac{p}{2} \right)} + \sin^{2}{\left(p \right)}\right)^{2}}{\left(4 \sin^{4}{\left(\frac{p}{2} \right)} + \sin^{2}{\left(p \right)}\right)^{2}} & \text{otherwise} \end{cases}}{294}\right)$$
/ 1 for p mod 2*pi = 0
|
| 2
|/ 2 \
/ 0 for p mod pi = 0 || sin (p) |
| ||-1 + ---------|
| 2 || 4/p\|
| sin (p) || 4*sin |-||
|------------------------ otherwise <\ \2//
| 2 |----------------- otherwise
$$\left(- \frac{\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\frac{\sin^{2}{\left(p \right)}}{\left(1 + \frac{\sin^{2}{\left(p \right)}}{4 \sin^{4}{\left(\frac{p}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{p}{2} \right)}} & \text{otherwise} \end{cases}}{294}\right) + \left(\frac{\begin{cases} 1 & \text{for}\: p \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sin^{2}{\left(p \right)}}{4 \sin^{4}{\left(\frac{p}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sin^{2}{\left(p \right)}}{4 \sin^{4}{\left(\frac{p}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}}{294}\right)$$
/ 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\ || 2
|| 4*cot |-| ||/ 2/p\\
<| \2/ <||-1 + cot |-||
|<-------------- otherwise otherwise |<\ \2// otherwise
|| 2 ||--------------- otherwise
||/ 2/p\\ || 2
|||1 + cot |-|| || / 2/p\\
||\ \2// || |1 + cot |-||
\\ \\ \ \2//
- ---------------------------------------------------- + ---------------------------------------------------------
294 294
$$\left(- \frac{\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{p}{2} \right)}}{\left(\cot^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}}{294}\right) + \left(\frac{\begin{cases} 1 & \text{for}\: p \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: p \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{p}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{p}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}}{294}\right)$$
/ 1 for p mod 2*pi = 0
|
| 2
|/ 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
$$\left(- \frac{\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\frac{4 \cos^{2}{\left(\frac{p}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{p}{2} \right)}}{\cos^{2}{\left(\frac{p}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \cos^{2}{\left(\frac{p}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}}{294}\right) + \left(\frac{\begin{cases} 1 & \text{for}\: p \bmod 2 \pi = 0 \\\frac{\left(\frac{\cos^{2}{\left(\frac{p}{2} \right)}}{\cos^{2}{\left(\frac{p}{2} - \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\cos^{2}{\left(\frac{p}{2} \right)}}{\cos^{2}{\left(\frac{p}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}}{294}\right)$$
/ 1 for p mod 2*pi = 0
|
| 2
|/ 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
$$\left(- \frac{\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\frac{4 \sec^{2}{\left(\frac{p}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{p}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{p}{2} \right)}}\right)^{2} \sec^{2}{\left(\frac{p}{2} \right)}} & \text{otherwise} \end{cases}}{294}\right) + \left(\frac{\begin{cases} 1 & \text{for}\: p \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{p}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{p}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sec^{2}{\left(\frac{p}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{p}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}}{294}\right)$$
/ 1 for p mod 2*pi = 0
|
| 2
|/ 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
$$\left(- \frac{\begin{cases} 0 & \text{for}\: p \bmod \pi = 0 \\\frac{4 \csc^{2}{\left(\frac{p}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{p}{2} \right)}}{\csc^{2}{\left(- \frac{p}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2} \csc^{2}{\left(- \frac{p}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}}{294}\right) + \left(\frac{\begin{cases} 1 & \text{for}\: p \bmod 2 \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{p}{2} \right)}}{\csc^{2}{\left(- \frac{p}{2} + \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\csc^{2}{\left(\frac{p}{2} \right)}}{\csc^{2}{\left(- \frac{p}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}}{294}\right)$$
-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))/294 + 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))/294