Господин Экзамен
25*cos(x)^2*sin(y)^2+25*cos(x)^2*cos(y)^2 если y=1
Решение
Вы ввели
[src]
2 2 2 2 25*cos (x)*sin (y) + 25*cos (x)*cos (y)
$$25 \sin^{2}{\left(y \right)} \cos^{2}{\left(x \right)} + 25 \cos^{2}{\left(x \right)} \cos^{2}{\left(y \right)}$$
25*cos(x)^2*sin(y)^2 + 25*cos(x)^2*cos(y)^2
Подстановка условия
[src]
25*cos(x)^2*sin(y)^2 + 25*cos(x)^2*cos(y)^2 при y = 1
подставляем
2 2 2 2 25*cos (x)*sin (y) + 25*cos (x)*cos (y)
$$25 \sin^{2}{\left(y \right)} \cos^{2}{\left(x \right)} + 25 \cos^{2}{\left(x \right)} \cos^{2}{\left(y \right)}$$
2 25*cos (x)
$$25 \cos^{2}{\left(x \right)}$$
переменные
y = 1
$$y = 1$$
2 25*cos (x)
$$25 \cos^{2}{\left(x \right)}$$
25*cos(x)^2
Собрать выражение
[src]
25 25*cos(2*x) -- + ----------- 2 2
$$\frac{25 \cos{\left(2 x \right)}}{2} + \frac{25}{2}$$
25/2 + 25*cos(2*x)/2
Тригонометрическая часть
[src]
2 25*cos (x)
$$25 \cos^{2}{\left(x \right)}$$
25 ------- 2 sec (x)
$$\frac{25}{\sec^{2}{\left(x \right)}}$$
2/ pi\
25*sin |x + --|
\ 2 /
$$25 \sin^{2}{\left(x + \frac{\pi}{2} \right)}$$
25 25*cos(2*x) -- + ----------- 2 2
$$\frac{25 \cos{\left(2 x \right)}}{2} + \frac{25}{2}$$
25
------------
2/pi \
csc |-- - x|
\2 /
$$\frac{25}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}$$
// 1 for x mod 2*pi = 0\ || | 25*|< 2 | ||cos (x) otherwise | \\ /
$$25 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)$$
2
/ 2/x\\
25*|1 - tan |-||
\ \2//
-----------------
2
/ 2/x\\
|1 + tan |-||
\ \2//
$$\frac{25 \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$
2 2 2 2/ pi\
25*cos (x)*cos (y) + 25*cos (x)*cos |y - --|
\ 2 /
$$25 \cos^{2}{\left(x \right)} \cos^{2}{\left(y \right)} + 25 \cos^{2}{\left(x \right)} \cos^{2}{\left(y - \frac{\pi}{2} \right)}$$
25 25 --------------- + --------------- 2 2 2 2 csc (y)*sec (x) sec (x)*sec (y)
$$\frac{25}{\sec^{2}{\left(x \right)} \sec^{2}{\left(y \right)}} + \frac{25}{\csc^{2}{\left(y \right)} \sec^{2}{\left(x \right)}}$$
25 25
--------------- + --------------------
2 2 2 2/ pi\
sec (x)*sec (y) sec (x)*sec |y - --|
\ 2 /
$$\frac{25}{\sec^{2}{\left(x \right)} \sec^{2}{\left(y - \frac{\pi}{2} \right)}} + \frac{25}{\sec^{2}{\left(x \right)} \sec^{2}{\left(y \right)}}$$
2 2 / /x\\ / /x\\ 4/x\ 25*|1 + tan|-|| *|-1 + tan|-|| *cos |-| \ \2// \ \2// \2/
$$25 \left(\tan{\left(\frac{x}{2} \right)} - 1\right)^{2} \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2} \cos^{4}{\left(\frac{x}{2} \right)}$$
25 25
--------------- + --------------------
2 2 2 2/pi \
sec (x)*sec (y) sec (x)*sec |-- - y|
\2 /
$$\frac{25}{\sec^{2}{\left(x \right)} \sec^{2}{\left(- y + \frac{\pi}{2} \right)}} + \frac{25}{\sec^{2}{\left(x \right)} \sec^{2}{\left(y \right)}}$$
2 2/ pi\ 2/ pi\ 2/ pi\
25*sin (y)*sin |x + --| + 25*sin |x + --|*sin |y + --|
\ 2 / \ 2 / \ 2 /
$$25 \sin^{2}{\left(y \right)} \sin^{2}{\left(x + \frac{\pi}{2} \right)} + 25 \sin^{2}{\left(x + \frac{\pi}{2} \right)} \sin^{2}{\left(y + \frac{\pi}{2} \right)}$$
25 25
-------------------- + -------------------------
2 2/pi \ 2/pi \ 2/pi \
csc (y)*csc |-- - x| csc |-- - x|*csc |-- - y|
\2 / \2 / \2 /
$$\frac{25}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)} \csc^{2}{\left(- y + \frac{\pi}{2} \right)}} + \frac{25}{\csc^{2}{\left(y \right)} \csc^{2}{\left(- x + \frac{\pi}{2} \right)}}$$
25 25
------------------------- + -------------------------
2 2/pi \ 2/pi \ 2/pi \
csc (pi - y)*csc |-- - x| csc |-- - x|*csc |-- - y|
\2 / \2 / \2 /
$$\frac{25}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)} \csc^{2}{\left(- y + \frac{\pi}{2} \right)}} + \frac{25}{\csc^{2}{\left(- y + \pi \right)} \csc^{2}{\left(- x + \frac{\pi}{2} \right)}}$$
4 2 2 / 2/x\\ / /x\\ / /x\\ 8/x\ 25*|1 - tan |-|| *|1 + tan|-|| *|-1 + tan|-|| *cos |-| \ \4// \ \2// \ \2// \4/
$$25 \left(- \tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4} \left(\tan{\left(\frac{x}{2} \right)} - 1\right)^{2} \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2} \cos^{8}{\left(\frac{x}{4} \right)}$$
2 2 / 1 1 \ / 1 1 \ 4/x\ 25*|1 + ------ - ------| *|-1 + ------ - ------| *cos |-| \ sin(x) tan(x)/ \ sin(x) tan(x)/ \2/
$$25 \left(-1 - \frac{1}{\tan{\left(x \right)}} + \frac{1}{\sin{\left(x \right)}}\right)^{2} \left(1 - \frac{1}{\tan{\left(x \right)}} + \frac{1}{\sin{\left(x \right)}}\right)^{2} \cos^{4}{\left(\frac{x}{2} \right)}$$
// 1 for x mod 2*pi = 0\ || | || 2 | ||/ 2/x\\ | |||-1 + cot |-|| | 25*|<\ \2// | ||--------------- otherwise | || 2 | || / 2/x\\ | || |1 + cot |-|| | \\ \ \2// /
$$25 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)$$
2 2 / 2/x\\ / 2/x\\ | 2*sin |-|| | 2*sin |-|| | \2/| | \2/| 4/pi x\ 25*|1 + ---------| *|-1 + ---------| *sin |-- + -| \ sin(x) / \ sin(x) / \2 2/
$$25 \left(\frac{2 \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} - 1\right)^{2} \left(\frac{2 \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} + 1\right)^{2} \sin^{4}{\left(\frac{x}{2} + \frac{\pi}{2} \right)}$$
4 2 2
/ 2/x\\ / /x\\ / /x\\
25*|1 - tan |-|| *|1 + tan|-|| *|-1 + tan|-||
\ \4// \ \2// \ \2//
----------------------------------------------
4
/ 2/x\\
|1 + tan |-||
\ \4//
$$\frac{25 \left(- \tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4} \left(\tan{\left(\frac{x}{2} \right)} - 1\right)^{2} \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4}}$$
2 2
25*(1 - cos(x) + sin(x)) *(-1 + cos(x) + sin(x))
-------------------------------------------------
2
/1 2 cos(2*x)\
|- + (1 - cos(x)) - --------|
\2 2 /
$$\frac{25 \left(\sin{\left(x \right)} - \cos{\left(x \right)} + 1\right)^{2} \left(\sin{\left(x \right)} + \cos{\left(x \right)} - 1\right)^{2}}{\left(\left(- \cos{\left(x \right)} + 1\right)^{2} - \frac{\cos{\left(2 x \right)}}{2} + \frac{1}{2}\right)^{2}}$$
2 2 / /x pi\\ / /x pi\\ | cos|- - --|| | cos|- - --|| | \2 2 /| | \2 2 /| 4/x\ 25*|1 + -----------| *|-1 + -----------| *cos |-| | /x\ | | /x\ | \2/ | cos|-| | | cos|-| | \ \2/ / \ \2/ /
$$25 \left(-1 + \frac{\cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos{\left(\frac{x}{2} \right)}}\right)^{2} \left(1 + \frac{\cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos{\left(\frac{x}{2} \right)}}\right)^{2} \cos^{4}{\left(\frac{x}{2} \right)}$$
/1 cos(2*x)\ /1 cos(2*y)\ /1 cos(2*x)\ /1 cos(2*y)\ 25*|- + --------|*|- + --------| + 25*|- + --------|*|- - --------| \2 2 / \2 2 / \2 2 / \2 2 /
$$25 \cdot \left(- \frac{\cos{\left(2 y \right)}}{2} + \frac{1}{2}\right) \left(\frac{\cos{\left(2 x \right)}}{2} + \frac{1}{2}\right) + 25 \left(\frac{\cos{\left(2 x \right)}}{2} + \frac{1}{2}\right) \left(\frac{\cos{\left(2 y \right)}}{2} + \frac{1}{2}\right)$$
2 2
/ /x\ \ / /x\ \
| sec|-| | | sec|-| |
| \2/ | | \2/ |
25*|1 + -----------| *|-1 + -----------|
| /x pi\| | /x pi\|
| sec|- - --|| | sec|- - --||
\ \2 2 // \ \2 2 //
-----------------------------------------
4/x\
sec |-|
\2/
$$\frac{25 \left(\frac{\sec{\left(\frac{x}{2} \right)}}{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1\right)^{2} \left(\frac{\sec{\left(\frac{x}{2} \right)}}{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}}{\sec^{4}{\left(\frac{x}{2} \right)}}$$
2 2
/ /pi x\\ / /pi x\\
| csc|-- - -|| | csc|-- - -||
| \2 2/| | \2 2/|
25*|1 + -----------| *|-1 + -----------|
| /x\ | | /x\ |
| csc|-| | | csc|-| |
\ \2/ / \ \2/ /
-----------------------------------------
4/pi x\
csc |-- - -|
\2 2/
$$\frac{25 \left(-1 + \frac{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc{\left(\frac{x}{2} \right)}}\right)^{2} \left(1 + \frac{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc{\left(\frac{x}{2} \right)}}\right)^{2}}{\csc^{4}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}$$
// x \
|| 1 for - mod 2*pi = 0|
2 2 || 2 |
/ /x\\ / /x\\ || |
25*|1 + tan|-|| *|-1 + tan|-|| *|< 4 |
\ \2// \ \2// ||/ 2/x\\ 8/x\ |
|||-1 + cot |-|| *sin |-| otherwise |
||\ \4// \4/ |
\\ /
$$25 \left(\tan{\left(\frac{x}{2} \right)} - 1\right)^{2} \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2} \left(\begin{cases} 1 & \text{for}\: \frac{x}{2} \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{4} \sin^{8}{\left(\frac{x}{4} \right)} & \text{otherwise} \end{cases}\right)$$
// x \
|| 1 for - mod 2*pi = 0|
|| 2 |
|| |
2 2 || 4 |
/ 1 \ / 1 \ ||/ 2/x\\ |
25*|1 + ------| *|-1 + ------| *|<|-1 + cot |-|| |
| /x\| | /x\| ||\ \4// |
| cot|-|| | cot|-|| ||--------------- otherwise |
\ \2// \ \2// || 4 |
|| / 2/x\\ |
|| |1 + cot |-|| |
\\ \ \4// /
$$25 \left(-1 + \frac{1}{\cot{\left(\frac{x}{2} \right)}}\right)^{2} \left(1 + \frac{1}{\cot{\left(\frac{x}{2} \right)}}\right)^{2} \left(\begin{cases} 1 & \text{for}\: \frac{x}{2} \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{4}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4}} & \text{otherwise} \end{cases}\right)$$
2 2
2 / /x\\ / /x\\
2 2 25*(1 + cos(x)) *|1 + tan|-|| *|-1 + tan|-|| *(1 - cos(2*y))
/ 2/x\\ / 2/y\\ 4/x\ 4/y\ \ \2// \ \2//
25*|1 - tan |-|| *|1 - tan |-|| *cos |-|*cos |-| + ------------------------------------------------------------
\ \2// \ \2// \2/ \2/ 8
$$25 \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \left(- \tan^{2}{\left(\frac{y}{2} \right)} + 1\right)^{2} \cos^{4}{\left(\frac{x}{2} \right)} \cos^{4}{\left(\frac{y}{2} \right)} + \frac{25 \cdot \left(- \cos{\left(2 y \right)} + 1\right) \left(\cos{\left(x \right)} + 1\right)^{2} \left(\tan{\left(\frac{x}{2} \right)} - 1\right)^{2} \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2}}{8}$$
/ 2 2 \ / 2 2 \ / 2 2 \ / 2 2 \ |1 cos (x) sin (x)| |1 cos (y) sin (y)| |1 cos (x) sin (x)| |1 sin (y) cos (y)| 25*|- + ------- - -------|*|- + ------- - -------| + 25*|- + ------- - -------|*|- + ------- - -------| \2 2 2 / \2 2 2 / \2 2 2 / \2 2 2 /
$$25 \left(- \frac{\sin^{2}{\left(x \right)}}{2} + \frac{\cos^{2}{\left(x \right)}}{2} + \frac{1}{2}\right) \left(- \frac{\sin^{2}{\left(y \right)}}{2} + \frac{\cos^{2}{\left(y \right)}}{2} + \frac{1}{2}\right) + 25 \left(- \frac{\sin^{2}{\left(x \right)}}{2} + \frac{\cos^{2}{\left(x \right)}}{2} + \frac{1}{2}\right) \left(\frac{\sin^{2}{\left(y \right)}}{2} - \frac{\cos^{2}{\left(y \right)}}{2} + \frac{1}{2}\right)$$
2 2
/ 2/y pi\\ / 2/x\\ 2 4/x\
2 2 25*|1 - cot |- + --|| *|1 - tan |-|| *(1 + sin(y)) *cos |-|
/ 2/x\\ / 2/y\\ 4/x\ 4/y\ \ \2 4 // \ \2// \2/
25*|1 - tan |-|| *|1 - tan |-|| *cos |-|*cos |-| + -----------------------------------------------------------
\ \2// \ \2// \2/ \2/ 4
$$25 \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \left(- \tan^{2}{\left(\frac{y}{2} \right)} + 1\right)^{2} \cos^{4}{\left(\frac{x}{2} \right)} \cos^{4}{\left(\frac{y}{2} \right)} + \frac{25 \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \left(- \cot^{2}{\left(\frac{y}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\sin{\left(y \right)} + 1\right)^{2} \cos^{4}{\left(\frac{x}{2} \right)}}{4}$$
2 2
/ 2/x\\ / 2/y pi\\ 4/x\
25*|-1 + cot |-|| *|-1 + tan |- + --|| *sin |-| 2 2
\ \2// \ \2 4 // \2/ / 2/x\\ / 2/y\\ 4/x\ 4/y\
----------------------------------------------- + 25*|-1 + cot |-|| *|-1 + cot |-|| *sin |-|*sin |-|
2 \ \2// \ \2// \2/ \2/
/ 2/y pi\\
|1 + tan |- + --||
\ \2 4 //
$$25 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2} \left(\cot^{2}{\left(\frac{y}{2} \right)} - 1\right)^{2} \sin^{4}{\left(\frac{x}{2} \right)} \sin^{4}{\left(\frac{y}{2} \right)} + \frac{25 \left(\tan^{2}{\left(\frac{y}{2} + \frac{\pi}{4} \right)} - 1\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2} \sin^{4}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{y}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
2 2 2
/ 2/x\\ / 2/y\\ / 2/x\\ 2/y\
25*|1 - tan |-|| *|1 - tan |-|| 100*|1 - tan |-|| *tan |-|
\ \2// \ \2// \ \2// \2/
-------------------------------- + -----------------------------
2 2 2 2
/ 2/x\\ / 2/y\\ / 2/x\\ / 2/y\\
|1 + tan |-|| *|1 + tan |-|| |1 + tan |-|| *|1 + tan |-||
\ \2// \ \2// \ \2// \ \2//
$$\frac{25 \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \left(- \tan^{2}{\left(\frac{y}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \left(\tan^{2}{\left(\frac{y}{2} \right)} + 1\right)^{2}} + \frac{100 \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \tan^{2}{\left(\frac{y}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \left(\tan^{2}{\left(\frac{y}{2} \right)} + 1\right)^{2}}$$
2 2
2/x pi\ 2/y pi\ / /x\\ / /x\\ 4/x\
400*tan |- + --|*tan |- + --| 25*|1 + tan|-|| *|-1 + tan|-|| *cos |-|*(1 - cos(2*y))
\2 4 / \2 4 / \ \2// \ \2// \2/
--------------------------------------- + ------------------------------------------------------
2 2 2
/ 2/x pi\\ / 2/y pi\\
|1 + tan |- + --|| *|1 + tan |- + --||
\ \2 4 // \ \2 4 //
$$\frac{25 \cdot \left(- \cos{\left(2 y \right)} + 1\right) \left(\tan{\left(\frac{x}{2} \right)} - 1\right)^{2} \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2} \cos^{4}{\left(\frac{x}{2} \right)}}{2} + \frac{400 \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} \tan^{2}{\left(\frac{y}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\tan^{2}{\left(\frac{y}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
2/y\ 2/x pi\ 2/x pi\ 2/y pi\
400*cot |-|*tan |- + --| 400*tan |- + --|*tan |- + --|
\2/ \2 4 / \2 4 / \2 4 /
---------------------------------- + ---------------------------------------
2 2 2 2
/ 2/y\\ / 2/x pi\\ / 2/x pi\\ / 2/y pi\\
|1 + cot |-|| *|1 + tan |- + --|| |1 + tan |- + --|| *|1 + tan |- + --||
\ \2// \ \2 4 // \ \2 4 // \ \2 4 //
$$\frac{400 \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} \cot^{2}{\left(\frac{y}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\cot^{2}{\left(\frac{y}{2} \right)} + 1\right)^{2}} + \frac{400 \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} \tan^{2}{\left(\frac{y}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\tan^{2}{\left(\frac{y}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
2/y\ 2/x pi\ 2/x pi\ 2/y pi\
400*tan |-|*tan |- + --| 400*tan |- + --|*tan |- + --|
\2/ \2 4 / \2 4 / \2 4 /
---------------------------------- + ---------------------------------------
2 2 2 2
/ 2/y\\ / 2/x pi\\ / 2/x pi\\ / 2/y pi\\
|1 + tan |-|| *|1 + tan |- + --|| |1 + tan |- + --|| *|1 + tan |- + --||
\ \2// \ \2 4 // \ \2 4 // \ \2 4 //
$$\frac{400 \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} \tan^{2}{\left(\frac{y}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\tan^{2}{\left(\frac{y}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} + \frac{400 \tan^{2}{\left(\frac{y}{2} \right)} \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{y}{2} \right)} + 1\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
2 2 2 2
/ 2/x\\ / 2/y\\ / 2/x\\ / 2/y pi\\
25*|-1 + cot |-|| *|-1 + cot |-|| 25*|-1 + cot |-|| *|-1 + tan |- + --||
\ \2// \ \2// \ \2// \ \2 4 //
---------------------------------- + ---------------------------------------
2 2 2 2
/ 2/x\\ / 2/y\\ / 2/x\\ / 2/y pi\\
|1 + cot |-|| *|1 + cot |-|| |1 + cot |-|| *|1 + tan |- + --||
\ \2// \ \2// \ \2// \ \2 4 //
$$\frac{25 \left(\tan^{2}{\left(\frac{y}{2} + \frac{\pi}{4} \right)} - 1\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{y}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + \frac{25 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2} \left(\cot^{2}{\left(\frac{y}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \left(\cot^{2}{\left(\frac{y}{2} \right)} + 1\right)^{2}}$$
// 0 for y mod pi = 0\ // 1 for x mod 2*pi = 0\ // 1 for x mod 2*pi = 0\ // 1 for y mod 2*pi = 0\ || | || | || | || | 25*|< 2 |*|< 2 | + 25*|< 2 |*|< 2 | ||sin (y) otherwise | ||cos (x) otherwise | ||cos (x) otherwise | ||cos (y) otherwise | \\ / \\ / \\ / \\ /
$$\left(25 \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\sin^{2}{\left(y \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(25 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: y \bmod 2 \pi = 0 \\\cos^{2}{\left(y \right)} & \text{otherwise} \end{cases}\right)\right)$$
2 2 2 2
/ 2/y pi\\ / 2/x\\ / 2/x\\ / 2/y\\
25*|1 - cot |- + --|| *|1 - tan |-|| 25*|1 - tan |-|| *|1 - tan |-||
\ \2 4 // \ \2// \ \2// \ \2//
------------------------------------- + --------------------------------
2 2 2 2
/ 2/y pi\\ / 2/x\\ / 2/x\\ / 2/y\\
|1 + cot |- + --|| *|1 + tan |-|| |1 + tan |-|| *|1 + tan |-||
\ \2 4 // \ \2// \ \2// \ \2//
$$\frac{25 \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \left(- \tan^{2}{\left(\frac{y}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \left(\tan^{2}{\left(\frac{y}{2} \right)} + 1\right)^{2}} + \frac{25 \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \left(- \cot^{2}{\left(\frac{y}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \left(\cot^{2}{\left(\frac{y}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
2 2 2
/ 1 \ / 1 \ / 1 \
25*|1 - -------| *|1 - -------| 100*|1 - -------|
| 2/x\| | 2/y\| | 2/x\|
| cot |-|| | cot |-|| | cot |-||
\ \2// \ \2// \ \2//
-------------------------------- + -------------------------------------
2 2 2 2
/ 1 \ / 1 \ / 1 \ / 1 \ 2/y\
|1 + -------| *|1 + -------| |1 + -------| *|1 + -------| *cot |-|
| 2/x\| | 2/y\| | 2/x\| | 2/y\| \2/
| cot |-|| | cot |-|| | cot |-|| | cot |-||
\ \2// \ \2// \ \2// \ \2//
$$\frac{25 \left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \left(1 - \frac{1}{\cot^{2}{\left(\frac{y}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \left(1 + \frac{1}{\cot^{2}{\left(\frac{y}{2} \right)}}\right)^{2}} + \frac{100 \left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \left(1 + \frac{1}{\cot^{2}{\left(\frac{y}{2} \right)}}\right)^{2} \cot^{2}{\left(\frac{y}{2} \right)}}$$
// 0 for y mod pi = 0\ // 1 for x mod 2*pi = 0\ // 1 for x mod 2*pi = 0\ // 1 for y mod 2*pi = 0\ || | || | || | || | 25*|< 2/ pi\ |*|< 2 | + 25*|< 2 |*|< 2 | ||cos |y - --| otherwise | ||cos (x) otherwise | ||cos (x) otherwise | ||cos (y) otherwise | \\ \ 2 / / \\ / \\ / \\ /
$$\left(25 \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\cos^{2}{\left(y - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(25 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: y \bmod 2 \pi = 0 \\\cos^{2}{\left(y \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for y mod pi = 0\ // 1 for x mod 2*pi = 0\ // 1 for x mod 2*pi = 0\ // 1 for y mod 2*pi = 0\ || | || | || | || | 25*|< 2 |*|< 2/ pi\ | + 25*|< 2/ pi\ |*|< 2/ pi\ | ||sin (y) otherwise | ||sin |x + --| otherwise | ||sin |x + --| otherwise | ||sin |y + --| otherwise | \\ / \\ \ 2 / / \\ \ 2 / / \\ \ 2 / /
$$\left(25 \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\sin^{2}{\left(y \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\sin^{2}{\left(x + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(25 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\sin^{2}{\left(x + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: y \bmod 2 \pi = 0 \\\sin^{2}{\left(y + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for y mod pi = 0\ // 1 for x mod 2*pi = 0\ // 1 for x mod 2*pi = 0\ // 1 for y mod 2*pi = 0\ || | || | || | || | || 1 | || 1 | || 1 | || 1 | 25*|<------------ otherwise |*|<------- otherwise | + 25*|<------- otherwise |*|<------- otherwise | || 2/ pi\ | || 2 | || 2 | || 2 | ||sec |y - --| | ||sec (x) | ||sec (x) | ||sec (y) | \\ \ 2 / / \\ / \\ / \\ /
$$\left(25 \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\frac{1}{\sec^{2}{\left(y - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{1}{\sec^{2}{\left(x \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(25 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{1}{\sec^{2}{\left(x \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: y \bmod 2 \pi = 0 \\\frac{1}{\sec^{2}{\left(y \right)}} & \text{otherwise} \end{cases}\right)\right)$$
// / 3*pi\ \
// 1 for x mod 2*pi = 0\ // 1 for y mod 2*pi = 0\ // 1 for x mod 2*pi = 0\ || 1 for |y + ----| mod 2*pi = 0|
|| | || | || | || \ 2 / |
25*|< 2 |*|< 2 | + 25*|< 2 |*|< |
||cos (x) otherwise | ||cos (y) otherwise | ||cos (x) otherwise | || 4/y\ 2/y\ |
\\ / \\ / \\ / ||- 4*cos |-| + 4*cos |-| otherwise |
\\ \2/ \2/ /
$$\left(25 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: y \bmod 2 \pi = 0 \\\cos^{2}{\left(y \right)} & \text{otherwise} \end{cases}\right)\right) + \left(25 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(y + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\- 4 \cos^{4}{\left(\frac{y}{2} \right)} + 4 \cos^{2}{\left(\frac{y}{2} \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for y mod pi = 0\ // 1 for x mod 2*pi = 0\ // 1 for x mod 2*pi = 0\ // 1 for y mod 2*pi = 0\ || | || | || | || | || 1 | || 1 | || 1 | || 1 | 25*|<------- otherwise |*|<------------ otherwise | + 25*|<------------ otherwise |*|<------------ otherwise | || 2 | || 2/pi \ | || 2/pi \ | || 2/pi \ | ||csc (y) | ||csc |-- - x| | ||csc |-- - x| | ||csc |-- - y| | \\ / \\ \2 / / \\ \2 / / \\ \2 / /
$$\left(25 \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\frac{1}{\csc^{2}{\left(y \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{1}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(25 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{1}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: y \bmod 2 \pi = 0 \\\frac{1}{\csc^{2}{\left(- y + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right)$$
2 2 2
/ 4/x\\ / 4/y\\ / 4/x\\
| 4*sin |-|| | 4*sin |-|| | 4*sin |-||
| \2/| | \2/| | \2/| 4/y\
25*|1 - ---------| *|1 - ---------| 400*|1 - ---------| *sin |-|
| 2 | | 2 | | 2 | \2/
\ sin (x) / \ sin (y) / \ sin (x) /
------------------------------------ + -----------------------------------------
2 2 2 2
/ 4/x\\ / 4/y\\ / 4/x\\ / 4/y\\
| 4*sin |-|| | 4*sin |-|| | 4*sin |-|| | 4*sin |-||
| \2/| | \2/| | \2/| | \2/| 2
|1 + ---------| *|1 + ---------| |1 + ---------| *|1 + ---------| *sin (y)
| 2 | | 2 | | 2 | | 2 |
\ sin (x) / \ sin (y) / \ sin (x) / \ sin (y) /
$$\frac{25 \left(- \frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{2} \left(- \frac{4 \sin^{4}{\left(\frac{y}{2} \right)}}{\sin^{2}{\left(y \right)}} + 1\right)^{2}}{\left(\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{2} \left(\frac{4 \sin^{4}{\left(\frac{y}{2} \right)}}{\sin^{2}{\left(y \right)}} + 1\right)^{2}} + \frac{400 \left(- \frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{2} \sin^{4}{\left(\frac{y}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{2} \left(\frac{4 \sin^{4}{\left(\frac{y}{2} \right)}}{\sin^{2}{\left(y \right)}} + 1\right)^{2} \sin^{2}{\left(y \right)}}$$
// / pi\ \ // / pi\ \ // / pi\ \
// 0 for y mod pi = 0\ || 0 for |x + --| mod pi = 0| || 0 for |x + --| mod pi = 0| || 0 for |y + --| mod pi = 0|
|| | || \ 2 / | || \ 2 / | || \ 2 / |
25*|< 2 |*|< | + 25*|< |*|< |
||sin (y) otherwise | || 2 2/x pi\ | || 2 2/x pi\ | || 2 2/y pi\ |
\\ / ||(1 + sin(x)) *cot |- + --| otherwise | ||(1 + sin(x)) *cot |- + --| otherwise | ||(1 + sin(y)) *cot |- + --| otherwise |
\\ \2 4 / / \\ \2 4 / / \\ \2 4 / /
$$\left(25 \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\sin^{2}{\left(y \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(x \right)} + 1\right)^{2} \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(25 \left(\begin{cases} 0 & \text{for}\: \left(x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(x \right)} + 1\right)^{2} \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(y + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(y \right)} + 1\right)^{2} \cot^{2}{\left(\frac{y}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for y mod pi = 0\ // 1 for x mod 2*pi = 0\ // 1 for x mod 2*pi = 0\ // 1 for y mod 2*pi = 0\ || | || | || | || | ||/ 0 for y mod pi = 0 | ||/ 1 for x mod 2*pi = 0 | ||/ 1 for x mod 2*pi = 0 | ||/ 1 for y mod 2*pi = 0 | 25*|<| |*|<| | + 25*|<| |*|<| | ||< 2 otherwise | ||< 2 otherwise | ||< 2 otherwise | ||< 2 otherwise | |||sin (y) otherwise | |||cos (x) otherwise | |||cos (x) otherwise | |||cos (y) otherwise | \\\ / \\\ / \\\ / \\\ /
$$\left(25 \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\sin^{2}{\left(y \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(25 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: y \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: y \bmod 2 \pi = 0 \\\cos^{2}{\left(y \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for y mod pi = 0\ // 1 for x mod 2*pi = 0\ // 1 for x mod 2*pi = 0\ // 1 for y mod 2*pi = 0\ || | || | || | || | || 2/y\ | || 2 | || 2 | || 2 | || 4*cot |-| | ||/ 2/x\\ | ||/ 2/x\\ | ||/ 2/y\\ | || \2/ | |||-1 + cot |-|| | |||-1 + cot |-|| | |||-1 + cot |-|| | 25*|<-------------- otherwise |*|<\ \2// | + 25*|<\ \2// |*|<\ \2// | || 2 | ||--------------- otherwise | ||--------------- otherwise | ||--------------- otherwise | ||/ 2/y\\ | || 2 | || 2 | || 2 | |||1 + cot |-|| | || / 2/x\\ | || / 2/x\\ | || / 2/y\\ | ||\ \2// | || |1 + cot |-|| | || |1 + cot |-|| | || |1 + cot |-|| | \\ / \\ \ \2// / \\ \ \2// / \\ \ \2// /
$$\left(25 \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{y}{2} \right)}}{\left(\cot^{2}{\left(\frac{y}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(25 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: y \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{y}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{y}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for y mod pi = 0\ // 1 for x mod 2*pi = 0\ // 1 for x mod 2*pi = 0\ // 1 for y mod 2*pi = 0\ || | || | || | || | || 2/y\ | || 2 | || 2 | || 2 | || 4*tan |-| | ||/ 2/x\\ | ||/ 2/x\\ | ||/ 2/y\\ | || \2/ | |||1 - tan |-|| | |||1 - tan |-|| | |||1 - tan |-|| | 25*|<-------------- otherwise |*|<\ \2// | + 25*|<\ \2// |*|<\ \2// | || 2 | ||-------------- otherwise | ||-------------- otherwise | ||-------------- otherwise | ||/ 2/y\\ | || 2 | || 2 | || 2 | |||1 + tan |-|| | ||/ 2/x\\ | ||/ 2/x\\ | ||/ 2/y\\ | ||\ \2// | |||1 + tan |-|| | |||1 + tan |-|| | |||1 + tan |-|| | \\ / \\\ \2// / \\\ \2// / \\\ \2// /
$$\left(25 \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\frac{4 \tan^{2}{\left(\frac{y}{2} \right)}}{\left(\tan^{2}{\left(\frac{y}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(25 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: y \bmod 2 \pi = 0 \\\frac{\left(- \tan^{2}{\left(\frac{y}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{y}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right)$$
2 2 2
/ 2/x pi\\ / 2/y pi\\ / 2/x pi\\
| cos |- - --|| | cos |- - --|| | cos |- - --||
| \2 2 /| | \2 2 /| | \2 2 /| 2/y pi\
25*|1 - ------------| *|1 - ------------| 100*|1 - ------------| *cos |- - --|
| 2/x\ | | 2/y\ | | 2/x\ | \2 2 /
| cos |-| | | cos |-| | | cos |-| |
\ \2/ / \ \2/ / \ \2/ /
------------------------------------------ + -----------------------------------------------
2 2 2 2
/ 2/x pi\\ / 2/y pi\\ / 2/x pi\\ / 2/y pi\\
| cos |- - --|| | cos |- - --|| | cos |- - --|| | cos |- - --||
| \2 2 /| | \2 2 /| | \2 2 /| | \2 2 /| 2/y\
|1 + ------------| *|1 + ------------| |1 + ------------| *|1 + ------------| *cos |-|
| 2/x\ | | 2/y\ | | 2/x\ | | 2/y\ | \2/
| cos |-| | | cos |-| | | cos |-| | | cos |-| |
\ \2/ / \ \2/ / \ \2/ / \ \2/ /
$$\frac{25 \left(1 - \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \left(1 - \frac{\cos^{2}{\left(\frac{y}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{y}{2} \right)}}\right)^{2}}{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \left(1 + \frac{\cos^{2}{\left(\frac{y}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{y}{2} \right)}}\right)^{2}} + \frac{100 \left(1 - \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \cos^{2}{\left(\frac{y}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \left(1 + \frac{\cos^{2}{\left(\frac{y}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{y}{2} \right)}}\right)^{2} \cos^{2}{\left(\frac{y}{2} \right)}}$$
2 2 2
/ 2/x\ \ / 2/y\ \ / 2/x\ \
| sec |-| | | sec |-| | | sec |-| |
| \2/ | | \2/ | | \2/ | 2/y\
25*|1 - ------------| *|1 - ------------| 100*|1 - ------------| *sec |-|
| 2/x pi\| | 2/y pi\| | 2/x pi\| \2/
| sec |- - --|| | sec |- - --|| | sec |- - --||
\ \2 2 // \ \2 2 // \ \2 2 //
------------------------------------------ + ----------------------------------------------------
2 2 2 2
/ 2/x\ \ / 2/y\ \ / 2/x\ \ / 2/y\ \
| sec |-| | | sec |-| | | sec |-| | | sec |-| |
| \2/ | | \2/ | | \2/ | | \2/ | 2/y pi\
|1 + ------------| *|1 + ------------| |1 + ------------| *|1 + ------------| *sec |- - --|
| 2/x pi\| | 2/y pi\| | 2/x pi\| | 2/y pi\| \2 2 /
| sec |- - --|| | sec |- - --|| | sec |- - --|| | sec |- - --||
\ \2 2 // \ \2 2 // \ \2 2 // \ \2 2 //
$$\frac{25 \left(- \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \left(- \frac{\sec^{2}{\left(\frac{y}{2} \right)}}{\sec^{2}{\left(\frac{y}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}}{\left(\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \left(\frac{\sec^{2}{\left(\frac{y}{2} \right)}}{\sec^{2}{\left(\frac{y}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}} + \frac{100 \left(- \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec^{2}{\left(\frac{y}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \left(\frac{\sec^{2}{\left(\frac{y}{2} \right)}}{\sec^{2}{\left(\frac{y}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec^{2}{\left(\frac{y}{2} - \frac{\pi}{2} \right)}}$$
2 2 2
/ 2/pi x\\ / 2/pi y\\ / 2/pi x\\
| csc |-- - -|| | csc |-- - -|| | csc |-- - -||
| \2 2/| | \2 2/| | \2 2/| 2/pi y\
25*|1 - ------------| *|1 - ------------| 100*|1 - ------------| *csc |-- - -|
| 2/x\ | | 2/y\ | | 2/x\ | \2 2/
| csc |-| | | csc |-| | | csc |-| |
\ \2/ / \ \2/ / \ \2/ /
------------------------------------------ + -----------------------------------------------
2 2 2 2
/ 2/pi x\\ / 2/pi y\\ / 2/pi x\\ / 2/pi y\\
| csc |-- - -|| | csc |-- - -|| | csc |-- - -|| | csc |-- - -||
| \2 2/| | \2 2/| | \2 2/| | \2 2/| 2/y\
|1 + ------------| *|1 + ------------| |1 + ------------| *|1 + ------------| *csc |-|
| 2/x\ | | 2/y\ | | 2/x\ | | 2/y\ | \2/
| csc |-| | | csc |-| | | csc |-| | | csc |-| |
\ \2/ / \ \2/ / \ \2/ / \ \2/ /
$$\frac{25 \left(1 - \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \left(1 - \frac{\csc^{2}{\left(- \frac{y}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{y}{2} \right)}}\right)^{2}}{\left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \left(1 + \frac{\csc^{2}{\left(- \frac{y}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{y}{2} \right)}}\right)^{2}} + \frac{100 \left(1 - \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \csc^{2}{\left(- \frac{y}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \left(1 + \frac{\csc^{2}{\left(- \frac{y}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{y}{2} \right)}}\right)^{2} \csc^{2}{\left(\frac{y}{2} \right)}}$$
// 1 for x mod 2*pi = 0\ // 1 for x mod 2*pi = 0\ // 1 for y mod 2*pi = 0\
|| | || | || |
// 0 for y mod pi = 0\ || 2 | || 2 | || 2 |
|| | ||/ 1 \ | ||/ 1 \ | ||/ 1 \ |
|| 4 | |||-1 + -------| | |||-1 + -------| | |||-1 + -------| |
||---------------------- otherwise | ||| 2/x\| | ||| 2/x\| | ||| 2/y\| |
|| 2 | ||| tan |-|| | ||| tan |-|| | ||| tan |-|| |
25*|
$$\left(25 \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\frac{4}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{y}{2} \right)}}\right)^{2} \tan^{2}{\left(\frac{y}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(25 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: y \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{y}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{y}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)\right)$$
// / 3*pi\ \
// 1 for x mod 2*pi = 0\ // 1 for y mod 2*pi = 0\ // 1 for x mod 2*pi = 0\ || 1 for |y + ----| mod 2*pi = 0|
|| | || | || | || \ 2 / |
|| 2 | || 2 | || 2 | || |
||/ 2/x\\ | ||/ 2/y\\ | ||/ 2/x\\ | || 2 |
|||-1 + cot |-|| | |||-1 + cot |-|| | |||-1 + cot |-|| | ||/ 2/y pi\\ |
25*|<\ \2// |*|<\ \2// | + 25*|<\ \2// |*|<|-1 + tan |- + --|| |
||--------------- otherwise | ||--------------- otherwise | ||--------------- otherwise | ||\ \2 4 // |
|| 2 | || 2 | || 2 | ||-------------------- otherwise |
|| / 2/x\\ | || / 2/y\\ | || / 2/x\\ | || 2 |
|| |1 + cot |-|| | || |1 + cot |-|| | || |1 + cot |-|| | ||/ 2/y pi\\ |
\\ \ \2// / \\ \ \2// / \\ \ \2// / |||1 + tan |- + --|| |
\\\ \2 4 // /
$$\left(25 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: y \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{y}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{y}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(25 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(y + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\left(\tan^{2}{\left(\frac{y}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{y}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right)$$
// / pi\ \ // / pi\ \ // / pi\ \
// 0 for y mod pi = 0\ || 0 for |x + --| mod pi = 0| || 0 for |x + --| mod pi = 0| || 0 for |y + --| mod pi = 0|
|| | || \ 2 / | || \ 2 / | || \ 2 / |
|| 2/y\ | || | || | || |
|| 4*cot |-| | || 2/x pi\ | || 2/x pi\ | || 2/y pi\ |
|| \2/ | || 4*cot |- + --| | || 4*cot |- + --| | || 4*cot |- + --| |
25*|<-------------- otherwise |*|< \2 4 / | + 25*|< \2 4 / |*|< \2 4 / |
|| 2 | ||------------------- otherwise | ||------------------- otherwise | ||------------------- otherwise |
||/ 2/y\\ | || 2 | || 2 | || 2 |
|||1 + cot |-|| | ||/ 2/x pi\\ | ||/ 2/x pi\\ | ||/ 2/y pi\\ |
||\ \2// | |||1 + cot |- + --|| | |||1 + cot |- + --|| | |||1 + cot |- + --|| |
\\ / ||\ \2 4 // | ||\ \2 4 // | ||\ \2 4 // |
\\ / \\ / \\ /
$$\left(25 \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{y}{2} \right)}}{\left(\cot^{2}{\left(\frac{y}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(25 \left(\begin{cases} 0 & \text{for}\: \left(x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(y + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{y}{2} + \frac{\pi}{4} \right)}}{\left(\cot^{2}{\left(\frac{y}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for y mod pi = 0\
|| | // 1 for x mod 2*pi = 0\ // 1 for x mod 2*pi = 0\ // 1 for y mod 2*pi = 0\
|| 2 | || | || | || |
|| sin (y) | || 2 | || 2 | || 2 |
||------------------------ otherwise | ||/ 2 4/x\\ | ||/ 2 4/x\\ | ||/ 2 4/y\\ |
|| 2 | |||sin (x) - 4*sin |-|| | |||sin (x) - 4*sin |-|| | |||sin (y) - 4*sin |-|| |
25*|
$$\left(25 \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\frac{\sin^{2}{\left(y \right)}}{\left(1 + \frac{\sin^{2}{\left(y \right)}}{4 \sin^{4}{\left(\frac{y}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{y}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(- 4 \sin^{4}{\left(\frac{x}{2} \right)} + \sin^{2}{\left(x \right)}\right)^{2}}{\left(4 \sin^{4}{\left(\frac{x}{2} \right)} + \sin^{2}{\left(x \right)}\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(25 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(- 4 \sin^{4}{\left(\frac{x}{2} \right)} + \sin^{2}{\left(x \right)}\right)^{2}}{\left(4 \sin^{4}{\left(\frac{x}{2} \right)} + \sin^{2}{\left(x \right)}\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: y \bmod 2 \pi = 0 \\\frac{\left(- 4 \sin^{4}{\left(\frac{y}{2} \right)} + \sin^{2}{\left(y \right)}\right)^{2}}{\left(4 \sin^{4}{\left(\frac{y}{2} \right)} + \sin^{2}{\left(y \right)}\right)^{2}} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for x mod 2*pi = 0\ // 1 for x mod 2*pi = 0\ // 1 for y mod 2*pi = 0\
|| | || | || |
// 0 for y mod pi = 0\ || 2 | || 2 | || 2 |
|| | ||/ 2 \ | ||/ 2 \ | ||/ 2 \ |
|| 2 | ||| sin (x) | | ||| sin (x) | | ||| sin (y) | |
|| sin (y) | |||-1 + ---------| | |||-1 + ---------| | |||-1 + ---------| |
||------------------------ otherwise | ||| 4/x\| | ||| 4/x\| | ||| 4/y\| |
|| 2 | ||| 4*sin |-|| | ||| 4*sin |-|| | ||| 4*sin |-|| |
25*|
$$\left(25 \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\frac{\sin^{2}{\left(y \right)}}{\left(1 + \frac{\sin^{2}{\left(y \right)}}{4 \sin^{4}{\left(\frac{y}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{y}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(25 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: y \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sin^{2}{\left(y \right)}}{4 \sin^{4}{\left(\frac{y}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sin^{2}{\left(y \right)}}{4 \sin^{4}{\left(\frac{y}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for y mod pi = 0\ // 1 for x mod 2*pi = 0\ // 1 for x mod 2*pi = 0\ // 1 for y mod 2*pi = 0\ || | || | || | || | ||/ 0 for y mod pi = 0 | ||/ 1 for x mod 2*pi = 0 | ||/ 1 for x mod 2*pi = 0 | ||/ 1 for y mod 2*pi = 0 | ||| | ||| | ||| | ||| | ||| 2/y\ | ||| 2 | ||| 2 | ||| 2 | ||| 4*cot |-| | |||/ 2/x\\ | |||/ 2/x\\ | |||/ 2/y\\ | 25*|<| \2/ |*|<||-1 + cot |-|| | + 25*|<||-1 + cot |-|| |*|<||-1 + cot |-|| | ||<-------------- otherwise otherwise | ||<\ \2// otherwise | ||<\ \2// otherwise | ||<\ \2// otherwise | ||| 2 | |||--------------- otherwise | |||--------------- otherwise | |||--------------- otherwise | |||/ 2/y\\ | ||| 2 | ||| 2 | ||| 2 | ||||1 + cot |-|| | ||| / 2/x\\ | ||| / 2/x\\ | ||| / 2/y\\ | |||\ \2// | ||| |1 + cot |-|| | ||| |1 + cot |-|| | ||| |1 + cot |-|| | \\\ / \\\ \ \2// / \\\ \ \2// / \\\ \ \2// /
$$\left(25 \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{y}{2} \right)}}{\left(\cot^{2}{\left(\frac{y}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(25 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: y \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: y \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{y}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{y}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for x mod 2*pi = 0\ // 1 for x mod 2*pi = 0\ // 1 for y mod 2*pi = 0\
|| | || | || |
// 0 for y mod pi = 0\ || 2 | || 2 | || 2 |
|| | ||/ 2/x\ \ | ||/ 2/x\ \ | ||/ 2/y\ \ |
|| 2/y\ | ||| cos |-| | | ||| cos |-| | | ||| cos |-| | |
|| 4*cos |-| | ||| \2/ | | ||| \2/ | | ||| \2/ | |
|| \2/ | |||-1 + ------------| | |||-1 + ------------| | |||-1 + ------------| |
||-------------------------------- otherwise | ||| 2/x pi\| | ||| 2/x pi\| | ||| 2/y pi\| |
|| 2 | ||| cos |- - --|| | ||| cos |- - --|| | ||| cos |- - --|| |
25*|
$$\left(25 \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\frac{4 \cos^{2}{\left(\frac{y}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{y}{2} \right)}}{\cos^{2}{\left(\frac{y}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \cos^{2}{\left(\frac{y}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(25 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: y \bmod 2 \pi = 0 \\\frac{\left(\frac{\cos^{2}{\left(\frac{y}{2} \right)}}{\cos^{2}{\left(\frac{y}{2} - \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\cos^{2}{\left(\frac{y}{2} \right)}}{\cos^{2}{\left(\frac{y}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for x mod 2*pi = 0\ // 1 for x mod 2*pi = 0\ // 1 for y mod 2*pi = 0\
|| | || | || |
// 0 for y mod pi = 0\ || 2 | || 2 | || 2 |
|| | ||/ 2/x pi\\ | ||/ 2/x pi\\ | ||/ 2/y pi\\ |
|| 2/y pi\ | ||| sec |- - --|| | ||| sec |- - --|| | ||| sec |- - --|| |
|| 4*sec |- - --| | ||| \2 2 /| | ||| \2 2 /| | ||| \2 2 /| |
|| \2 2 / | |||-1 + ------------| | |||-1 + ------------| | |||-1 + ------------| |
||--------------------------- otherwise | ||| 2/x\ | | ||| 2/x\ | | ||| 2/y\ | |
|| 2 | ||| sec |-| | | ||| sec |-| | | ||| sec |-| | |
25*|
$$\left(25 \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\frac{4 \sec^{2}{\left(\frac{y}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{y}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{y}{2} \right)}}\right)^{2} \sec^{2}{\left(\frac{y}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(25 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: y \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{y}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{y}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sec^{2}{\left(\frac{y}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{y}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for x mod 2*pi = 0\ // 1 for x mod 2*pi = 0\ // 1 for y mod 2*pi = 0\
|| | || | || |
// 0 for y mod pi = 0\ || 2 | || 2 | || 2 |
|| | ||/ 2/x\ \ | ||/ 2/x\ \ | ||/ 2/y\ \ |
|| 2/y\ | ||| csc |-| | | ||| csc |-| | | ||| csc |-| | |
|| 4*csc |-| | ||| \2/ | | ||| \2/ | | ||| \2/ | |
|| \2/ | |||-1 + ------------| | |||-1 + ------------| | |||-1 + ------------| |
||-------------------------------- otherwise | ||| 2/pi x\| | ||| 2/pi x\| | ||| 2/pi y\| |
|| 2 | ||| csc |-- - -|| | ||| csc |-- - -|| | ||| csc |-- - -|| |
25*|
$$\left(25 \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\frac{4 \csc^{2}{\left(\frac{y}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{y}{2} \right)}}{\csc^{2}{\left(- \frac{y}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2} \csc^{2}{\left(- \frac{y}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(25 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: y \bmod 2 \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{y}{2} \right)}}{\csc^{2}{\left(- \frac{y}{2} + \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\csc^{2}{\left(\frac{y}{2} \right)}}{\csc^{2}{\left(- \frac{y}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right)$$
25*Piecewise((0, Mod(y = pi, 0)), (4*csc(y/2)^2/((1 + csc(y/2)^2/csc(pi/2 - y/2)^2)^2*csc(pi/2 - y/2)^2), True))*Piecewise((1, Mod(x = 2*pi, 0)), ((-1 + csc(x/2)^2/csc(pi/2 - x/2)^2)^2/(1 + csc(x/2)^2/csc(pi/2 - x/2)^2)^2, True)) + 25*Piecewise((1, Mod(x = 2*pi, 0)), ((-1 + csc(x/2)^2/csc(pi/2 - x/2)^2)^2/(1 + csc(x/2)^2/csc(pi/2 - x/2)^2)^2, True))*Piecewise((1, Mod(y = 2*pi, 0)), ((-1 + csc(y/2)^2/csc(pi/2 - y/2)^2)^2/(1 + csc(y/2)^2/csc(pi/2 - y/2)^2)^2, True))
Численный ответ
[src]
25.0*cos(x)^2*cos(y)^2 + 25.0*cos(x)^2*sin(y)^2
25.0*cos(x)^2*cos(y)^2 + 25.0*cos(x)^2*sin(y)^2
Степени
[src]
2
/ I*x -I*x\ 2
2 2 |e e | / -I*y I*y\
/ I*x -I*x\ / I*y -I*y\ 25*|---- + -----| *\- e + e /
|e e | |e e | \ 2 2 /
25*|---- + -----| *|---- + -----| - ------------------------------------
\ 2 2 / \ 2 2 / 4
$$25 \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{2} \left(\frac{e^{i y}}{2} + \frac{e^{- i y}}{2}\right)^{2} - \frac{25 \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{2} \left(e^{i y} - e^{- i y}\right)^{2}}{4}$$
25*(exp(i*x)/2 + exp(-i*x)/2)^2*(exp(i*y)/2 + exp(-i*y)/2)^2 - 25*(exp(i*x)/2 + exp(-i*x)/2)^2*(-exp(-i*y) + exp(i*y))^2/4
Объединение рациональных выражений
[src]
2 / 2 2 \ 25*cos (x)*\cos (y) + sin (y)/
$$25 \left(\sin^{2}{\left(y \right)} + \cos^{2}{\left(y \right)}\right) \cos^{2}{\left(x \right)}$$
25*cos(x)^2*(cos(y)^2 + sin(y)^2)
Комбинаторика
[src]
2 / 2 2 \ 25*cos (x)*\cos (y) + sin (y)/
$$25 \left(\sin^{2}{\left(y \right)} + \cos^{2}{\left(y \right)}\right) \cos^{2}{\left(x \right)}$$
25*cos(x)^2*(cos(y)^2 + sin(y)^2)