Господин Экзамен
Общий знаменатель atan((4*sin(x)^2-2)/2)/2
Решение
Вы ввели
[src]
/ 2 \
|4*sin (x) - 2|
atan|-------------|
\ 2 /
-------------------
2
$$\frac{\operatorname{atan}{\left(\frac{4 \sin^{2}{\left(x \right)} - 2}{2} \right)}}{2}$$
atan((4*sin(x)^2 - 1*2)/2)/2
Общее упрощение
[src]
-atan(cos(2*x))
----------------
2
$$- \frac{\operatorname{atan}{\left(\cos{\left(2 x \right)} \right)}}{2}$$
-atan(cos(2*x))/2
Разложение дроби
[src]
atan(-1 + 2*sin(x)^2)/2
$$\frac{\operatorname{atan}{\left(2 \sin^{2}{\left(x \right)} - 1 \right)}}{2}$$
/ 2 \
atan\-1 + 2*sin (x)/
--------------------
2
Объединение рациональных выражений
[src]
/ 2 \
atan\-1 + 2*sin (x)/
--------------------
2
$$\frac{\operatorname{atan}{\left(2 \sin^{2}{\left(x \right)} - 1 \right)}}{2}$$
atan(-1 + 2*sin(x)^2)/2
Общий знаменатель
[src]
/ 2 \
atan\-1 + 2*sin (x)/
--------------------
2
$$\frac{\operatorname{atan}{\left(2 \sin^{2}{\left(x \right)} - 1 \right)}}{2}$$
atan(-1 + 2*sin(x)^2)/2
Комбинаторика
[src]
/ 2 \
atan\-1 + 2*sin (x)/
--------------------
2
$$\frac{\operatorname{atan}{\left(2 \sin^{2}{\left(x \right)} - 1 \right)}}{2}$$
atan(-1 + 2*sin(x)^2)/2
Рациональный знаменатель
[src]
/ 2 \
atan\-1 + 2*sin (x)/
--------------------
2
$$\frac{\operatorname{atan}{\left(2 \sin^{2}{\left(x \right)} - 1 \right)}}{2}$$
atan(-1 + 2*sin(x)^2)/2
Степени
[src]
/ 2 \
atan\-1 + 2*sin (x)/
--------------------
2
$$\frac{\operatorname{atan}{\left(2 \sin^{2}{\left(x \right)} - 1 \right)}}{2}$$
/ 2\
| / -I*x I*x\ |
| \- e + e / |
-atan|1 + -----------------|
\ 2 /
-----------------------------
2
$$- \frac{\operatorname{atan}{\left(\frac{\left(e^{i x} - e^{- i x}\right)^{2}}{2} + 1 \right)}}{2}$$
-atan(1 + (-exp(-i*x) + exp(i*x))^2/2)/2
Собрать выражение
[src]
-atan(cos(2*x))
----------------
2
$$- \frac{\operatorname{atan}{\left(\cos{\left(2 x \right)} \right)}}{2}$$
-atan(cos(2*x))/2
Тригонометрическая часть
[src]
-atan(cos(2*x))
----------------
2
$$- \frac{\operatorname{atan}{\left(\cos{\left(2 x \right)} \right)}}{2}$$
/ 2 \
atan\-1 + 2*sin (x)/
--------------------
2
$$\frac{\operatorname{atan}{\left(2 \sin^{2}{\left(x \right)} - 1 \right)}}{2}$$
/ 1 \
-acot|--------|
\cos(2*x)/
----------------
2
$$- \frac{\operatorname{acot}{\left(\frac{1}{\cos{\left(2 x \right)}} \right)}}{2}$$
/ 1 \
-atan|--------|
\sec(2*x)/
----------------
2
$$- \frac{\operatorname{atan}{\left(\frac{1}{\sec{\left(2 x \right)}} \right)}}{2}$$
/ 2 \
-atan|1 - -------|
| 2 |
\ csc (x)/
-------------------
2
$$- \frac{\operatorname{atan}{\left(1 - \frac{2}{\csc^{2}{\left(x \right)}} \right)}}{2}$$
/ /pi \\
-atan|sin|-- + 2*x||
\ \2 //
---------------------
2
$$- \frac{\operatorname{atan}{\left(\sin{\left(2 x + \frac{\pi}{2} \right)} \right)}}{2}$$
/ 2/ pi\\
atan|-1 + 2*cos |x - --||
\ \ 2 //
-------------------------
2
$$\frac{\operatorname{atan}{\left(2 \cos^{2}{\left(x - \frac{\pi}{2} \right)} - 1 \right)}}{2}$$
/ 2 \
-atan|1 - ------------|
| 2 |
\ csc (pi - x)/
------------------------
2
$$- \frac{\operatorname{atan}{\left(1 - \frac{2}{\csc^{2}{\left(- x + \pi \right)}} \right)}}{2}$$
/ 2 2 \
atan\sin (x) - cos (x)/
-----------------------
2
$$\frac{\operatorname{atan}{\left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)} \right)}}{2}$$
/ 1 \
-atan|-------------|
| /pi \|
|csc|-- - 2*x||
\ \2 //
---------------------
2
$$- \frac{\operatorname{atan}{\left(\frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}} \right)}}{2}$$
/ 2 \
-atan|1 - ------------|
| 2/ pi\|
| sec |x - --||
\ \ 2 //
------------------------
2
$$- \frac{\operatorname{atan}{\left(1 - \frac{2}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} \right)}}{2}$$
/ 2 \
-atan|1 - ------------|
| 2/pi \|
| sec |-- - x||
\ \2 //
------------------------
2
$$- \frac{\operatorname{atan}{\left(1 - \frac{2}{\sec^{2}{\left(- x + \frac{\pi}{2} \right)}} \right)}}{2}$$
/ 2 \
|1 - tan (x)|
-atan|-----------|
| 2 |
\1 + tan (x)/
-------------------
2
$$- \frac{\operatorname{atan}{\left(\frac{- \tan^{2}{\left(x \right)} + 1}{\tan^{2}{\left(x \right)} + 1} \right)}}{2}$$
// 1 for x mod pi = 0\
-atan|< |
\\cos(2*x) otherwise /
-----------------------------------
2
$$- \frac{\operatorname{atan}{\left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\cos{\left(2 x \right)} & \text{otherwise} \end{cases} \right)}}{2}$$
/ 2/x\ \
| 8*cot |-| |
| \2/ |
-atan|1 - --------------|
| 2|
| / 2/x\\ |
| |1 + cot |-|| |
\ \ \2// /
--------------------------
2
$$- \frac{\operatorname{atan}{\left(1 - \frac{8 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} \right)}}{2}$$
/ 2/x\ \
| 8*tan |-| |
| \2/ |
-atan|1 - --------------|
| 2|
| / 2/x\\ |
| |1 + tan |-|| |
\ \ \2// /
--------------------------
2
$$- \frac{\operatorname{atan}{\left(1 - \frac{8 \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} \right)}}{2}$$
/ 8 \
-atan|1 - ----------------------|
| 2 |
| / 1 \ 2/x\|
| |1 + -------| *cot |-||
| | 2/x\| \2/|
| | cot |-|| |
\ \ \2// /
----------------------------------
2
$$- \frac{\operatorname{atan}{\left(1 - \frac{8}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \cot^{2}{\left(\frac{x}{2} \right)}} \right)}}{2}$$
/ 2 \
| / 2/x pi\\ 2|
| |1 - cot |- + --|| *(1 + sin(x)) |
| \ \2 4 // |
atan|-1 + ---------------------------------|
\ 2 /
--------------------------------------------
2
$$\frac{\operatorname{atan}{\left(\frac{\left(- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\sin{\left(x \right)} + 1\right)^{2}}{2} - 1 \right)}}{2}$$
/ // 0 for x mod pi = 0\\
| || ||
atan|-1 + 2*|< 2 ||
| ||sin (x) otherwise ||
\ \\ //
-----------------------------------------
2
$$\frac{\operatorname{atan}{\left(\left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) - 1 \right)}}{2}$$
/ // 0 for x mod pi = 0\\
| || ||
| || 1 ||
atan|-1 + 2*|<------- otherwise ||
| || 2 ||
| ||csc (x) ||
\ \\ //
-----------------------------------------
2
$$\frac{\operatorname{atan}{\left(\left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\csc^{2}{\left(x \right)}} & \text{otherwise} \end{cases}\right)\right) - 1 \right)}}{2}$$
// 1 for x mod pi = 0\
|| |
|| 2 |
-atan|<-1 + cot (x) |
||------------ otherwise |
|| 2 |
\\1 + cot (x) /
---------------------------------------
2
$$- \frac{\operatorname{atan}{\left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases} \right)}}{2}$$
/ // 0 for x mod pi = 0\\
| || ||
atan|-1 + 2*|< 2/ pi\ ||
| ||cos |x - --| otherwise ||
\ \\ \ 2 / //
----------------------------------------------
2
$$\frac{\operatorname{atan}{\left(\left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\cos^{2}{\left(x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) - 1 \right)}}{2}$$
/ 2\
| / 2/x pi\\ |
| 2*|-1 + tan |- + --|| |
| \ \2 4 // |
atan|-1 + ----------------------|
| 2 |
| / 2/x pi\\ |
| |1 + tan |- + --|| |
\ \ \2 4 // /
---------------------------------
2
$$\frac{\operatorname{atan}{\left(\frac{2 \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} - 1 \right)}}{2}$$
/ 2\
| / 2/x pi\\ |
| 2*|1 - cot |- + --|| |
| \ \2 4 // |
atan|-1 + ---------------------|
| 2 |
| / 2/x pi\\ |
| |1 + cot |- + --|| |
\ \ \2 4 // /
--------------------------------
2
$$\frac{\operatorname{atan}{\left(\frac{2 \left(- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} - 1 \right)}}{2}$$
/ 2 4/x\ \
| 32*sin (x)*sin |-| |
| \2/ |
-atan|1 - ----------------------|
| 2|
| / 2 4/x\\ |
| |sin (x) + 4*sin |-|| |
\ \ \2// /
----------------------------------
2
$$- \frac{\operatorname{atan}{\left(- \frac{32 \sin^{4}{\left(\frac{x}{2} \right)} \sin^{2}{\left(x \right)}}{\left(4 \sin^{4}{\left(\frac{x}{2} \right)} + \sin^{2}{\left(x \right)}\right)^{2}} + 1 \right)}}{2}$$
/ // 0 for x mod pi = 0\\
| || ||
| || 1 ||
atan|-1 + 2*|<------------ otherwise ||
| || 2/ pi\ ||
| ||sec |x - --| ||
\ \\ \ 2 / //
----------------------------------------------
2
$$\frac{\operatorname{atan}{\left(\left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - 1 \right)}}{2}$$
/ 4/x\ \
| 32*sin |-| |
| \2/ |
-atan|1 - ------------------------|
| 2 |
| / 4/x\\ |
| | 4*sin |-|| |
| | \2/| 2 |
| |1 + ---------| *sin (x)|
| | 2 | |
\ \ sin (x) / /
------------------------------------
2
$$- \frac{\operatorname{atan}{\left(1 - \frac{32 \sin^{4}{\left(\frac{x}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{2} \sin^{2}{\left(x \right)}} \right)}}{2}$$
/ // 0 for x mod pi = 0\\
| || ||
| || 2/x\ ||
| || 4*cot |-| ||
| || \2/ ||
atan|-1 + 2*|<-------------- otherwise ||
| || 2 ||
| ||/ 2/x\\ ||
| |||1 + cot |-|| ||
| ||\ \2// ||
\ \\ //
------------------------------------------------
2
$$\frac{\operatorname{atan}{\left(\left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) - 1 \right)}}{2}$$
/ // 0 for x mod pi = 0\\
| || ||
| || 2/x\ ||
| || 4*tan |-| ||
| || \2/ ||
atan|-1 + 2*|<-------------- otherwise ||
| || 2 ||
| ||/ 2/x\\ ||
| |||1 + tan |-|| ||
| ||\ \2// ||
\ \\ //
------------------------------------------------
2
$$\frac{\operatorname{atan}{\left(\left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) - 1 \right)}}{2}$$
/ // 0 for x mod pi = 0\\
| || ||
| ||/ 0 for x mod pi = 0 ||
atan|-1 + 2*|<| ||
| ||< 2 otherwise ||
| |||sin (x) otherwise ||
\ \\\ //
------------------------------------------------------------
2
$$\frac{\operatorname{atan}{\left(\left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) - 1 \right)}}{2}$$
/ 2/x\ \
| 8*sec |-| |
| \2/ |
-atan|1 - --------------------------------|
| 2 |
| / 2/x\ \ |
| | sec |-| | |
| | \2/ | 2/x pi\|
| |1 + ------------| *sec |- - --||
| | 2/x pi\| \2 2 /|
| | sec |- - --|| |
\ \ \2 2 // /
--------------------------------------------
2
$$- \frac{\operatorname{atan}{\left(1 - \frac{8 \sec^{2}{\left(\frac{x}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} \right)}}{2}$$
/ 2/x pi\ \
| 8*cos |- - --| |
| \2 2 / |
-atan|1 - ---------------------------|
| 2 |
| / 2/x pi\\ |
| | cos |- - --|| |
| | \2 2 /| 2/x\|
| |1 + ------------| *cos |-||
| | 2/x\ | \2/|
| | cos |-| | |
\ \ \2/ / /
---------------------------------------
2
$$- \frac{\operatorname{atan}{\left(1 - \frac{8 \cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \cos^{2}{\left(\frac{x}{2} \right)}} \right)}}{2}$$
/ 2/pi x\ \
| 8*csc |-- - -| |
| \2 2/ |
-atan|1 - ---------------------------|
| 2 |
| / 2/pi x\\ |
| | csc |-- - -|| |
| | \2 2/| 2/x\|
| |1 + ------------| *csc |-||
| | 2/x\ | \2/|
| | csc |-| | |
\ \ \2/ / /
---------------------------------------
2
$$- \frac{\operatorname{atan}{\left(1 - \frac{8 \csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \csc^{2}{\left(\frac{x}{2} \right)}} \right)}}{2}$$
/ // / 3*pi\ \\
| || 1 for |x + ----| mod 2*pi = 0||
| || \ 2 / ||
atan|-1 + 2*|< ||
| || 4/x\ 2/x\ ||
| ||- 4*cos |-| + 4*cos |-| otherwise ||
\ \\ \2/ \2/ //
--------------------------------------------------------------------
2
$$\frac{\operatorname{atan}{\left(\left(2 \left(\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\- 4 \cos^{4}{\left(\frac{x}{2} \right)} + 4 \cos^{2}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right)\right) - 1 \right)}}{2}$$
/ // 0 for x mod pi = 0\\
| || ||
| || 4 ||
| ||---------------------- otherwise ||
| || 2 ||
atan|-1 + 2*|
$$\frac{\operatorname{atan}{\left(\left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \tan^{2}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - 1 \right)}}{2}$$
/ // 0 for x mod pi = 0\\
| || ||
| || 2 ||
| || sin (x) ||
| ||------------------------ otherwise ||
| || 2 ||
atan|-1 + 2*|
$$\frac{\operatorname{atan}{\left(\left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{\sin^{2}{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - 1 \right)}}{2}$$
/ // 0 for x mod pi = 0\\
| || ||
| ||/ 0 for x mod pi = 0 ||
| ||| ||
| ||| 2/x\ ||
| ||| 4*cot |-| ||
atan|-1 + 2*|<| \2/ ||
| ||<-------------- otherwise otherwise ||
| ||| 2 ||
| |||/ 2/x\\ ||
| ||||1 + cot |-|| ||
| |||\ \2// ||
\ \\\ //
-------------------------------------------------------------------
2
$$\frac{\operatorname{atan}{\left(\left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) - 1 \right)}}{2}$$
/ // / 3*pi\ \\
| || 1 for |x + ----| mod 2*pi = 0||
| || \ 2 / ||
| || ||
| || 2 ||
| ||/ 2/x pi\\ ||
atan|-1 + 2*|<|-1 + tan |- + --|| ||
| ||\ \2 4 // ||
| ||-------------------- otherwise ||
| || 2 ||
| ||/ 2/x pi\\ ||
| |||1 + tan |- + --|| ||
\ \\\ \2 4 // //
-----------------------------------------------------------------
2
$$\frac{\operatorname{atan}{\left(\left(2 \left(\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) - 1 \right)}}{2}$$
/ // 0 for x mod pi = 0\\
| || ||
| || 2/x pi\ ||
| || 4*sec |- - --| ||
| || \2 2 / ||
| ||--------------------------- otherwise ||
| || 2 ||
atan|-1 + 2*|
$$\frac{\operatorname{atan}{\left(\left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \sec^{2}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - 1 \right)}}{2}$$
/ // 0 for x mod pi = 0\\
| || ||
| || 2/x\ ||
| || 4*cos |-| ||
| || \2/ ||
| ||-------------------------------- otherwise ||
| || 2 ||
atan|-1 + 2*|
$$\frac{\operatorname{atan}{\left(\left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \cos^{2}{\left(\frac{x}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - 1 \right)}}{2}$$
/ // 0 for x mod pi = 0\\
| || ||
| || 2/x\ ||
| || 4*csc |-| ||
| || \2/ ||
| ||-------------------------------- otherwise ||
| || 2 ||
atan|-1 + 2*|
$$\frac{\operatorname{atan}{\left(\left(2 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \csc^{2}{\left(\frac{x}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2} \csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - 1 \right)}}{2}$$
atan(-1 + 2*Piecewise((0, Mod(x = pi, 0)), (4*csc(x/2)^2/((1 + csc(x/2)^2/csc(pi/2 - x/2)^2)^2*csc(pi/2 - x/2)^2), True)))/2