Господин Экзамен
sin(pi+t)^2+sin(pi-t)^2 если t=-1/4
Решение
Вы ввели
[src]
2 2 sin (pi + t) + sin (pi - t)
$$\sin^{2}{\left(- t + \pi \right)} + \sin^{2}{\left(t + \pi \right)}$$
sin(pi + t)^2 + sin(pi - t)^2
Подстановка условия
[src]
sin(pi + t)^2 + sin(pi - t)^2 при t = -1/4
подставляем
2 2 sin (pi + t) + sin (pi - t)
$$\sin^{2}{\left(- t + \pi \right)} + \sin^{2}{\left(t + \pi \right)}$$
2 2*sin (t)
$$2 \sin^{2}{\left(t \right)}$$
переменные
t = -1/4
$$t = - \frac{1}{4}$$
2 2*sin ((-1/4))
$$2 \sin^{2}{\left((-1/4) \right)}$$
2 2*sin (-1/4)
$$2 \sin^{2}{\left(- \frac{1}{4} \right)}$$
2 2*sin (1/4)
$$2 \sin^{2}{\left(\frac{1}{4} \right)}$$
2*sin(1/4)^2
Степени
[src]
2 2*sin (t)
$$2 \sin^{2}{\left(t \right)}$$
2 2
/ I*(t - pi) I*(pi - t)\ / I*(-pi - t) I*(pi + t)\
\- e + e / \- e + e /
- ------------------------------ - -------------------------------
4 4
$$- \frac{\left(e^{i \left(- t + \pi\right)} - e^{i \left(t - \pi\right)}\right)^{2}}{4} - \frac{\left(- e^{i \left(- t - \pi\right)} + e^{i \left(t + \pi\right)}\right)^{2}}{4}$$
-(-exp(i*(t - pi)) + exp(i*(pi - t)))^2/4 - (-exp(i*(-pi - t)) + exp(i*(pi + t)))^2/4
Общий знаменатель
[src]
2 2 sin (t) + sin (pi + t)
$$\sin^{2}{\left(t \right)} + \sin^{2}{\left(t + \pi \right)}$$
sin(t)^2 + sin(pi + t)^2
Тригонометрическая часть
[src]
2 2*sin (t)
$$2 \sin^{2}{\left(t \right)}$$
1 - cos(2*t)
$$- \cos{\left(2 t \right)} + 1$$
2 ------- 2 csc (t)
$$\frac{2}{\csc^{2}{\left(t \right)}}$$
1
1 - --------
sec(2*t)
$$1 - \frac{1}{\sec{\left(2 t \right)}}$$
2 ------------ 2 csc (pi - t)
$$\frac{2}{\csc^{2}{\left(- t + \pi \right)}}$$
2/ pi\
2*cos |t - --|
\ 2 /
$$2 \cos^{2}{\left(t - \frac{\pi}{2} \right)}$$
/pi \
1 - sin|-- + 2*t|
\2 /
$$- \sin{\left(2 t + \frac{\pi}{2} \right)} + 1$$
2
------------
2/ pi\
sec |t - --|
\ 2 /
$$\frac{2}{\sec^{2}{\left(t - \frac{\pi}{2} \right)}}$$
2
------------
2/pi \
sec |-- - t|
\2 /
$$\frac{2}{\sec^{2}{\left(- t + \frac{\pi}{2} \right)}}$$
1
1 - -------------
/pi \
csc|-- - 2*t|
\2 /
$$1 - \frac{1}{\csc{\left(- 2 t + \frac{\pi}{2} \right)}}$$
2 2 1 + sin (t) - cos (t)
$$\sin^{2}{\left(t \right)} - \cos^{2}{\left(t \right)} + 1$$
2
1 - tan (t)
1 - -----------
2
1 + tan (t)
$$- \frac{- \tan^{2}{\left(t \right)} + 1}{\tan^{2}{\left(t \right)} + 1} + 1$$
4/t\ 2/t\
- 8*cos |-| + 8*cos |-|
\2/ \2/
$$- 8 \cos^{4}{\left(\frac{t}{2} \right)} + 8 \cos^{2}{\left(\frac{t}{2} \right)}$$
2/t\
8*tan |-|
\2/
--------------
2
/ 2/t\\
|1 + tan |-||
\ \2//
$$\frac{8 \tan^{2}{\left(\frac{t}{2} \right)}}{\left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}}$$
2/t\
8*cot |-|
\2/
--------------
2
/ 2/t\\
|1 + cot |-||
\ \2//
$$\frac{8 \cot^{2}{\left(\frac{t}{2} \right)}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}}$$
8
----------------------
2
/ 1 \ 2/t\
|1 + -------| *cot |-|
| 2/t\| \2/
| cot |-||
\ \2//
$$\frac{8}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \cot^{2}{\left(\frac{t}{2} \right)}}$$
// 0 for t mod pi = 0\ || | 2*|< 2 | ||sin (t) otherwise | \\ /
$$2 \left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\sin^{2}{\left(t \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for t mod pi = 0\
1 - |< |
\\cos(2*t) otherwise /
$$\left(- \begin{cases} 1 & \text{for}\: t \bmod \pi = 0 \\\cos{\left(2 t \right)} & \text{otherwise} \end{cases}\right) + 1$$
2
/ 2/t pi\\ 2
|1 - cot |- + --|| *(1 + sin(t))
\ \2 4 //
---------------------------------
2
$$\frac{\left(- \cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\sin{\left(t \right)} + 1\right)^{2}}{2}$$
// 0 for t mod pi = 0\ || | || 1 | 2*|<------- otherwise | || 2 | ||csc (t) | \\ /
$$2 \left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{1}{\csc^{2}{\left(t \right)}} & \text{otherwise} \end{cases}\right)$$
// 0 for t mod pi = 0\ || | 2*|< 2/ pi\ | ||cos |t - --| otherwise | \\ \ 2 / /
$$2 \left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\cos^{2}{\left(t - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)$$
// 0 for t mod pi = 0\ || | || 1 | 2*|<------------ otherwise | || 2/ pi\ | ||sec |t - --| | \\ \ 2 / /
$$2 \left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{1}{\sec^{2}{\left(t - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
2
/ 2/t pi\\
2*|-1 + tan |- + --||
\ \2 4 //
----------------------
2
/ 2/t pi\\
|1 + tan |- + --||
\ \2 4 //
$$\frac{2 \left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
2 4/t\
32*sin (t)*sin |-|
\2/
----------------------
2
/ 2 4/t\\
|sin (t) + 4*sin |-||
\ \2//
$$\frac{32 \sin^{4}{\left(\frac{t}{2} \right)} \sin^{2}{\left(t \right)}}{\left(4 \sin^{4}{\left(\frac{t}{2} \right)} + \sin^{2}{\left(t \right)}\right)^{2}}$$
2
/ 2/t pi\\
2*|1 - cot |- + --||
\ \2 4 //
---------------------
2
/ 2/t pi\\
|1 + cot |- + --||
\ \2 4 //
$$\frac{2 \left(- \cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}{\left(\cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
4/t\
32*sin |-|
\2/
------------------------
2
/ 4/t\\
| 4*sin |-||
| \2/| 2
|1 + ---------| *sin (t)
| 2 |
\ sin (t) /
$$\frac{32 \sin^{4}{\left(\frac{t}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{t}{2} \right)}}{\sin^{2}{\left(t \right)}} + 1\right)^{2} \sin^{2}{\left(t \right)}}$$
// 1 for t mod pi = 0\
|| |
|| 2 |
1 - |<-1 + cot (t) |
||------------ otherwise |
|| 2 |
\\1 + cot (t) /
$$\left(- \begin{cases} 1 & \text{for}\: t \bmod \pi = 0 \\\frac{\cot^{2}{\left(t \right)} - 1}{\cot^{2}{\left(t \right)} + 1} & \text{otherwise} \end{cases}\right) + 1$$
// 0 for t mod pi = 0\ || | || 2/t\ | || 4*tan |-| | || \2/ | 2*|<-------------- otherwise | || 2 | ||/ 2/t\\ | |||1 + tan |-|| | ||\ \2// | \\ /
$$2 \left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{4 \tan^{2}{\left(\frac{t}{2} \right)}}{\left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// 0 for t mod pi = 0\ || | || 2/t\ | || 4*cot |-| | || \2/ | 2*|<-------------- otherwise | || 2 | ||/ 2/t\\ | |||1 + cot |-|| | ||\ \2// | \\ /
$$2 \left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{t}{2} \right)}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// 0 for t mod pi = 0\ || | ||/ 0 for t mod pi = 0 | 2*|<| | ||< 2 otherwise | |||sin (t) otherwise | \\\ /
$$2 \left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\sin^{2}{\left(t \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
// / 3*pi\ \ || 1 for |t + ----| mod 2*pi = 0| || \ 2 / | 2*|< | || 4/t\ 2/t\ | ||- 4*cos |-| + 4*cos |-| otherwise | \\ \2/ \2/ /
$$2 \left(\begin{cases} 1 & \text{for}\: \left(t + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\- 4 \cos^{4}{\left(\frac{t}{2} \right)} + 4 \cos^{2}{\left(\frac{t}{2} \right)} & \text{otherwise} \end{cases}\right)$$
// 0 for t mod pi = 0\
|| |
|| 4 |
||---------------------- otherwise |
|| 2 |
2*|
$$2 \left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{4}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \tan^{2}{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
2/t\
8*sec |-|
\2/
--------------------------------
2
/ 2/t\ \
| sec |-| |
| \2/ | 2/t pi\
|1 + ------------| *sec |- - --|
| 2/t pi\| \2 2 /
| sec |- - --||
\ \2 2 //
$$\frac{8 \sec^{2}{\left(\frac{t}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{t}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}$$
2/t pi\
8*cos |- - --|
\2 2 /
---------------------------
2
/ 2/t pi\\
| cos |- - --||
| \2 2 /| 2/t\
|1 + ------------| *cos |-|
| 2/t\ | \2/
| cos |-| |
\ \2/ /
$$\frac{8 \cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \cos^{2}{\left(\frac{t}{2} \right)}}$$
2/pi t\
8*csc |-- - -|
\2 2/
---------------------------
2
/ 2/pi t\\
| csc |-- - -||
| \2 2/| 2/t\
|1 + ------------| *csc |-|
| 2/t\ | \2/
| csc |-| |
\ \2/ /
$$\frac{8 \csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \csc^{2}{\left(\frac{t}{2} \right)}}$$
// 0 for t mod pi = 0\
|| |
|| 2 |
|| sin (t) |
||------------------------ otherwise |
|| 2 |
2*|
$$2 \left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{\sin^{2}{\left(t \right)}}{\left(1 + \frac{\sin^{2}{\left(t \right)}}{4 \sin^{4}{\left(\frac{t}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// 0 for t mod pi = 0\ || | ||/ 0 for t mod pi = 0 | ||| | ||| 2/t\ | ||| 4*cot |-| | 2*|<| \2/ | ||<-------------- otherwise otherwise | ||| 2 | |||/ 2/t\\ | ||||1 + cot |-|| | |||\ \2// | \\\ /
$$2 \left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{t}{2} \right)}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
// / 3*pi\ \ || 1 for |t + ----| mod 2*pi = 0| || \ 2 / | || | || 2 | ||/ 2/t pi\\ | 2*|<|-1 + tan |- + --|| | ||\ \2 4 // | ||-------------------- otherwise | || 2 | ||/ 2/t pi\\ | |||1 + tan |- + --|| | \\\ \2 4 // /
$$2 \left(\begin{cases} 1 & \text{for}\: \left(t + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)$$
// 0 for t mod pi = 0\
|| |
|| 2/t\ |
|| 4*cos |-| |
|| \2/ |
||-------------------------------- otherwise |
|| 2 |
2*|
$$2 \left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{4 \cos^{2}{\left(\frac{t}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{t}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// 0 for t mod pi = 0\
|| |
|| 2/t pi\ |
|| 4*sec |- - --| |
|| \2 2 / |
||--------------------------- otherwise |
|| 2 |
2*|
$$2 \left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{4 \sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} \right)}}\right)^{2} \sec^{2}{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// 0 for t mod pi = 0\
|| |
|| 2/t\ |
|| 4*csc |-| |
|| \2/ |
||-------------------------------- otherwise |
|| 2 |
2*|
$$2 \left(\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{4 \csc^{2}{\left(\frac{t}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{t}{2} \right)}}{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2} \csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
2*Piecewise((0, Mod(t = pi, 0)), (4*csc(t/2)^2/((1 + csc(t/2)^2/csc(pi/2 - t/2)^2)^2*csc(pi/2 - t/2)^2), True))