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