Господин Экзамен
1/(sin(x)^2)-1 если x=-1/4
Решение
Вы ввели
[src]
1
1*------- - 1
2
sin (x)
$$\left(-1\right) 1 + 1 \cdot \frac{1}{\sin^{2}{\left(x \right)}}$$
1/sin(x)^2 - 1*1
Разложение дроби
[src]
-1 + sin(x)^(-2)
$$-1 + \frac{1}{\sin^{2}{\left(x \right)}}$$
1
-1 + -------
2
sin (x)
Подстановка условия
[src]
1/sin(x)^2 - 1*1 при x = -1/4
подставляем
1
1*------- - 1
2
sin (x)
$$\left(-1\right) 1 + 1 \cdot \frac{1}{\sin^{2}{\left(x \right)}}$$
1 ------- 2 tan (x)
$$\frac{1}{\tan^{2}{\left(x \right)}}$$
переменные
x = -1/4
$$x = - \frac{1}{4}$$
1 ------------ 2 tan ((-1/4))
$$\frac{1}{\tan^{2}{\left((-1/4) \right)}}$$
1 ---------- 2 tan (-1/4)
$$\frac{1}{\tan^{2}{\left(- \frac{1}{4} \right)}}$$
1 --------- 2 tan (1/4)
$$\frac{1}{\tan^{2}{\left(\frac{1}{4} \right)}}$$
tan(1/4)^(-2)
Собрать выражение
[src]
2 -1 + csc (x)
$$\csc^{2}{\left(x \right)} - 1$$
1
-1 + -------
2
sin (x)
$$-1 + \frac{1}{\sin^{2}{\left(x \right)}}$$
-1 + sin(x)^(-2)
Общий знаменатель
[src]
1
-1 + -------
2
sin (x)
$$-1 + \frac{1}{\sin^{2}{\left(x \right)}}$$
-1 + sin(x)^(-2)
Степени
[src]
1
-1 + -------
2
sin (x)
$$-1 + \frac{1}{\sin^{2}{\left(x \right)}}$$
4
-1 - -----------------
2
/ -I*x I*x\
\- e + e /
$$-1 - \frac{4}{\left(e^{i x} - e^{- i x}\right)^{2}}$$
-1 - 4/(-exp(-i*x) + exp(i*x))^2
Рациональный знаменатель
[src]
1
-1 + -------
2
sin (x)
$$-1 + \frac{1}{\sin^{2}{\left(x \right)}}$$
2
1 - sin (x)
-----------
2
sin (x)
$$\frac{- \sin^{2}{\left(x \right)} + 1}{\sin^{2}{\left(x \right)}}$$
(1 - sin(x)^2)/sin(x)^2
Объединение рациональных выражений
[src]
2
1 - sin (x)
-----------
2
sin (x)
$$\frac{- \sin^{2}{\left(x \right)} + 1}{\sin^{2}{\left(x \right)}}$$
(1 - sin(x)^2)/sin(x)^2
Комбинаторика
[src]
-(1 + sin(x))*(-1 + sin(x))
----------------------------
2
sin (x)
$$- \frac{\left(\sin{\left(x \right)} - 1\right) \left(\sin{\left(x \right)} + 1\right)}{\sin^{2}{\left(x \right)}}$$
-(1 + sin(x))*(-1 + sin(x))/sin(x)^2
Тригонометрическая часть
[src]
2 cot (x)
$$\cot^{2}{\left(x \right)}$$
2 -1 + csc (x)
$$\csc^{2}{\left(x \right)} - 1$$
1 ------- 2 tan (x)
$$\frac{1}{\tan^{2}{\left(x \right)}}$$
2 -1 + csc (pi - x)
$$\csc^{2}{\left(- x + \pi \right)} - 1$$
1
-1 + -------
2
sin (x)
$$-1 + \frac{1}{\sin^{2}{\left(x \right)}}$$
2/ pi\
-1 + sec |x - --|
\ 2 /
$$\sec^{2}{\left(x - \frac{\pi}{2} \right)} - 1$$
2/pi \
-1 + sec |-- - x|
\2 /
$$\sec^{2}{\left(- x + \frac{\pi}{2} \right)} - 1$$
1
-1 + ------------
2/ pi\
cos |x - --|
\ 2 /
$$-1 + \frac{1}{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}$$
2
sin (2*x)
---------
4
4*sin (x)
$$\frac{\sin^{2}{\left(2 x \right)}}{4 \sin^{4}{\left(x \right)}}$$
2/ pi\
sec |x - --|
\ 2 /
------------
2
sec (x)
$$\frac{\sec^{2}{\left(x - \frac{\pi}{2} \right)}}{\sec^{2}{\left(x \right)}}$$
2
cos (x)
------------
2/ pi\
cos |x - --|
\ 2 /
$$\frac{\cos^{2}{\left(x \right)}}{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}$$
2
csc (x)
------------
2/pi \
csc |-- - x|
\2 /
$$\frac{\csc^{2}{\left(x \right)}}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}$$
1
-1 + ---------------------
2 2/x\
(1 + cos(x)) *tan |-|
\2/
$$-1 + \frac{1}{\left(\cos{\left(x \right)} + 1\right)^{2} \tan^{2}{\left(\frac{x}{2} \right)}}$$
2
/ 2/x\\
|1 + tan |-||
\ \2//
-1 + --------------
2/x\
4*tan |-|
\2/
$$\frac{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{4 \tan^{2}{\left(\frac{x}{2} \right)}} - 1$$
2
/ 2/x\\
|1 + cot |-||
\ \2//
-1 + --------------
2/x\
4*cot |-|
\2/
$$\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{4 \cot^{2}{\left(\frac{x}{2} \right)}} - 1$$
2
/ 1 \ 2/x\
|1 + -------| *cot |-|
| 2/x\| \2/
| cot |-||
\ \2//
-1 + ----------------------
4
$$\frac{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \cot^{2}{\left(\frac{x}{2} \right)}}{4} - 1$$
4
-1 + ---------------------------------
2
/ 2/x pi\\ 2
|1 - cot |- + --|| *(1 + sin(x))
\ \2 4 //
$$-1 + \frac{4}{\left(- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\sin{\left(x \right)} + 1\right)^{2}}$$
// zoo for x mod pi = 0\
|| |
-1 + |< 2 |
||csc (x) otherwise |
\\ /
$$\left(\begin{cases} \tilde{\infty} & \text{for}\: x \bmod \pi = 0 \\\csc^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) - 1$$
// zoo for x mod pi = 0\
|| |
|| 1 |
-1 + |<------- otherwise |
|| 2 |
||sin (x) |
\\ /
$$\left(\begin{cases} \tilde{\infty} & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\sin^{2}{\left(x \right)}} & \text{otherwise} \end{cases}\right) - 1$$
// zoo for x mod pi = 0\
|| |
-1 + |< 2/ pi\ |
||sec |x - --| otherwise |
\\ \ 2 / /
$$\left(\begin{cases} \tilde{\infty} & \text{for}\: x \bmod \pi = 0 \\\sec^{2}{\left(x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) - 1$$
2
/ 2/x pi\\
|1 + tan |- + --||
\ \2 4 //
-1 + --------------------
2
/ 2/x pi\\
|-1 + tan |- + --||
\ \2 4 //
$$-1 + \frac{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}$$
4/x\
sin |-| 2
1 \2/ sin (x)
- - + ------- + ----------
2 2 4/x\
sin (x) 16*sin |-|
\2/
$$\frac{\sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} - \frac{1}{2} + \frac{\sin^{2}{\left(x \right)}}{16 \sin^{4}{\left(\frac{x}{2} \right)}}$$
2
/ 2/x pi\\
|1 + cot |- + --||
\ \2 4 //
-1 + -------------------
2
/ 2/x pi\\
|1 - cot |- + --||
\ \2 4 //
$$-1 + \frac{\left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}{\left(- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
// zoo for x mod pi = 0\
|| |
|| 1 |
-1 + |<------------ otherwise |
|| 2/ pi\ |
||cos |x - --| |
\\ \ 2 / /
$$\left(\begin{cases} \tilde{\infty} & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\cos^{2}{\left(x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) - 1$$
2
/ 4/x\\
| 4*sin |-||
| \2/| 2
|1 + ---------| *sin (x)
| 2 |
\ sin (x) /
-1 + ------------------------
4/x\
16*sin |-|
\2/
$$\frac{\left(\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{2} \sin^{2}{\left(x \right)}}{16 \sin^{4}{\left(\frac{x}{2} \right)}} - 1$$
// zoo for x mod pi = 0\
|| |
|| 2/x\ |
|| tan |-| |
-1 + |< \2/ |
||-------------- otherwise |
|| 2 |
||(-1 + cos(x)) |
\\ /
$$\left(\begin{cases} \tilde{\infty} & \text{for}\: x \bmod \pi = 0 \\\frac{\tan^{2}{\left(\frac{x}{2} \right)}}{\left(\cos{\left(x \right)} - 1\right)^{2}} & \text{otherwise} \end{cases}\right) - 1$$
// zoo for x mod pi = 0\
|| |
|| 2 |
||/ 2/x\\ |
|||1 + tan |-|| |
-1 + |<\ \2// |
||-------------- otherwise |
|| 2/x\ |
|| 4*tan |-| |
|| \2/ |
\\ /
$$\left(\begin{cases} \tilde{\infty} & \text{for}\: x \bmod \pi = 0 \\\frac{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{4 \tan^{2}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) - 1$$
// zoo for x mod pi = 0\
|| |
|| 2 |
||/ 2/x\\ |
|||1 + cot |-|| |
-1 + |<\ \2// |
||-------------- otherwise |
|| 2/x\ |
|| 4*cot |-| |
|| \2/ |
\\ /
$$\left(\begin{cases} \tilde{\infty} & \text{for}\: x \bmod \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{4 \cot^{2}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) - 1$$
2
/ 2/x\ \
| sec |-| |
| \2/ | 2/x pi\
|1 + ------------| *sec |- - --|
| 2/x pi\| \2 2 /
| sec |- - --||
\ \2 2 //
-1 + --------------------------------
2/x\
4*sec |-|
\2/
$$\frac{\left(\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{4 \sec^{2}{\left(\frac{x}{2} \right)}} - 1$$
2
/ 2/x pi\\
| cos |- - --||
| \2 2 /| 2/x\
|1 + ------------| *cos |-|
| 2/x\ | \2/
| cos |-| |
\ \2/ /
-1 + ---------------------------
2/x pi\
4*cos |- - --|
\2 2 /
$$\frac{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \cos^{2}{\left(\frac{x}{2} \right)}}{4 \cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1$$
2
/ 2/pi x\\
| csc |-- - -||
| \2 2/| 2/x\
|1 + ------------| *csc |-|
| 2/x\ | \2/
| csc |-| |
\ \2/ /
-1 + ---------------------------
2/pi x\
4*csc |-- - -|
\2 2/
$$\frac{\left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \csc^{2}{\left(\frac{x}{2} \right)}}{4 \csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - 1$$
// zoo for x mod pi = 0\
|| |
|| 2/x\ |
|| -2*sin |-| |
-1 + |< \2/ |
||--------------------- otherwise |
|| 2 |
||(-1 + cos(x))*sin (x) |
\\ /
$$\left(\begin{cases} \tilde{\infty} & \text{for}\: x \bmod \pi = 0 \\- \frac{2 \sin^{2}{\left(\frac{x}{2} \right)}}{\left(\cos{\left(x \right)} - 1\right) \sin^{2}{\left(x \right)}} & \text{otherwise} \end{cases}\right) - 1$$
// zoo for x mod pi = 0\
|| |
|| 2 |
||/ 1 \ 2/x\ |
|||1 + -------| *tan |-| |
-1 + |<| 2/x\| \2/ |
||| tan |-|| |
||\ \2// |
||---------------------- otherwise |
|| 4 |
\\ /
$$\left(\begin{cases} \tilde{\infty} & \text{for}\: x \bmod \pi = 0 \\\frac{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \tan^{2}{\left(\frac{x}{2} \right)}}{4} & \text{otherwise} \end{cases}\right) - 1$$
// / 3*pi\ \
|| 1 for |x + ----| mod 2*pi = 0|
|| \ 2 / |
|| |
|| 1 |
||-1 + ------- |
-1 + |< 2/x\ |
|| sin |-| |
|| \2/ |
||------------ otherwise |
|| 4/x\ |
|| 4*cos |-| |
\\ \2/ /
$$\left(\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\sin^{2}{\left(\frac{x}{2} \right)}}}{4 \cos^{4}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) - 1$$
// zoo for x mod pi = 0\
|| |
||/ zoo for x mod pi = 0 |
||| |
-1 + |<| 1 |
||<------- otherwise otherwise |
||| 2 |
|||sin (x) |
\\\ /
$$\left(\begin{cases} \tilde{\infty} & \text{for}\: x \bmod \pi = 0 \\\begin{cases} \tilde{\infty} & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\sin^{2}{\left(x \right)}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) - 1$$
// zoo for x mod pi = 0\
|| |
|| 4/x\ |
|| sin |-| 2 |
-1 + |<1 \2/ sin (x) |
||- + ------- + ---------- otherwise |
||2 2 4/x\ |
|| sin (x) 16*sin |-| |
\\ \2/ /
$$\left(\begin{cases} \tilde{\infty} & \text{for}\: x \bmod \pi = 0 \\\frac{\sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + \frac{1}{2} + \frac{\sin^{2}{\left(x \right)}}{16 \sin^{4}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) - 1$$
// zoo for x mod pi = 0\
|| |
|| 2 |
||/ 2 \ |
||| sin (x) | 4/x\ |
|||1 + ---------| *sin |-| |
-1 + |<| 4/x\| \2/ |
||| 4*sin |-|| |
||\ \2// |
||------------------------ otherwise |
|| 2 |
|| sin (x) |
\\ /
$$\left(\begin{cases} \tilde{\infty} & \text{for}\: x \bmod \pi = 0 \\\frac{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} & \text{otherwise} \end{cases}\right) - 1$$
// / 3*pi\ \
|| 1 for |x + ----| mod 2*pi = 0|
|| \ 2 / |
|| |
|| 2 |
||/ 2/x pi\\ |
-1 + |<|1 + tan |- + --|| |
||\ \2 4 // |
||-------------------- otherwise |
|| 2 |
||/ 2/x pi\\ |
|||-1 + tan |- + --|| |
\\\ \2 4 // /
$$\left(\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}} & \text{otherwise} \end{cases}\right) - 1$$
// zoo for x mod pi = 0\
|| |
||/ zoo for x mod pi = 0 |
||| |
||| 2 |
|||/ 2/x\\ |
-1 + |<||1 + cot |-|| |
||<\ \2// otherwise |
|||-------------- otherwise |
||| 2/x\ |
||| 4*cot |-| |
||| \2/ |
\\\ /
$$\left(\begin{cases} \tilde{\infty} & \text{for}\: x \bmod \pi = 0 \\\begin{cases} \tilde{\infty} & \text{for}\: x \bmod \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{4 \cot^{2}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) - 1$$
// zoo for x mod pi = 0\
|| |
|| 2 |
||/ 2/x pi\\ |
||| sec |- - --|| |
||| \2 2 /| 2/x\ |
|||1 + ------------| *sec |-| |
-1 + |<| 2/x\ | \2/ |
||| sec |-| | |
||\ \2/ / |
||--------------------------- otherwise |
|| 2/x pi\ |
|| 4*sec |- - --| |
|| \2 2 / |
\\ /
$$\left(\begin{cases} \tilde{\infty} & \text{for}\: x \bmod \pi = 0 \\\frac{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \sec^{2}{\left(\frac{x}{2} \right)}}{4 \sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) - 1$$
// zoo for x mod pi = 0\
|| |
|| 2 |
||/ 2/x\ \ |
||| cos |-| | |
||| \2/ | 2/x pi\ |
|||1 + ------------| *cos |- - --| |
-1 + |<| 2/x pi\| \2 2 / |
||| cos |- - --|| |
||\ \2 2 // |
||-------------------------------- otherwise |
|| 2/x\ |
|| 4*cos |-| |
|| \2/ |
\\ /
$$\left(\begin{cases} \tilde{\infty} & \text{for}\: x \bmod \pi = 0 \\\frac{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{4 \cos^{2}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) - 1$$
// zoo for x mod pi = 0\
|| |
|| 2 |
||/ 2/x\ \ |
||| csc |-| | |
||| \2/ | 2/pi x\ |
|||1 + ------------| *csc |-- - -| |
-1 + |<| 2/pi x\| \2 2/ |
||| csc |-- - -|| |
||\ \2 2// |
||-------------------------------- otherwise |
|| 2/x\ |
|| 4*csc |-| |
|| \2/ |
\\ /
$$\left(\begin{cases} \tilde{\infty} & \text{for}\: x \bmod \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2} \csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{4 \csc^{2}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) - 1$$
-1 + Piecewise((±oo, Mod(x = pi, 0)), ((1 + csc(x/2)^2/csc(pi/2 - x/2)^2)^2*csc(pi/2 - x/2)^2/(4*csc(x/2)^2), True))