Господин Экзамен
1-cos(t)^2 если t=4
Решение
Подстановка условия
[src]
1 - cos(t)^2 при t = 4
подставляем
2 1 - cos (t)
$$- \cos^{2}{\left(t \right)} + 1$$
2 sin (t)
$$\sin^{2}{\left(t \right)}$$
переменные
t = 4
$$t = 4$$
2 sin ((4))
$$\sin^{2}{\left((4) \right)}$$
2 sin (4)
$$\sin^{2}{\left(4 \right)}$$
sin(4)^2
Собрать выражение
[src]
1 cos(2*t) - - -------- 2 2
$$- \frac{\cos{\left(2 t \right)}}{2} + \frac{1}{2}$$
1/2 - cos(2*t)/2
Комбинаторика
[src]
-(1 + cos(t))*(-1 + cos(t))
$$- \left(\cos{\left(t \right)} - 1\right) \left(\cos{\left(t \right)} + 1\right)$$
-(1 + cos(t))*(-1 + cos(t))
Степени
[src]
2
/ I*t -I*t\
|e e |
1 - |---- + -----|
\ 2 2 /
$$- \left(\frac{e^{i t}}{2} + \frac{e^{- i t}}{2}\right)^{2} + 1$$
1 - (exp(i*t)/2 + exp(-i*t)/2)^2
Тригонометрическая часть
[src]
2 sin (t)
$$\sin^{2}{\left(t \right)}$$
1 ------- 2 csc (t)
$$\frac{1}{\csc^{2}{\left(t \right)}}$$
1
1 - -------
2
sec (t)
$$1 - \frac{1}{\sec^{2}{\left(t \right)}}$$
2/ pi\
cos |t - --|
\ 2 /
$$\cos^{2}{\left(t - \frac{\pi}{2} \right)}$$
2/ pi\
1 - sin |t + --|
\ 2 /
$$- \sin^{2}{\left(t + \frac{\pi}{2} \right)} + 1$$
1 cos(2*t) - - -------- 2 2
$$- \frac{\cos{\left(2 t \right)}}{2} + \frac{1}{2}$$
1
------------
2/ pi\
sec |t - --|
\ 2 /
$$\frac{1}{\sec^{2}{\left(t - \frac{\pi}{2} \right)}}$$
1
1 - ------------
2/pi \
csc |-- - t|
\2 /
$$1 - \frac{1}{\csc^{2}{\left(- t + \frac{\pi}{2} \right)}}$$
4/t\ 2/t\
- 4*cos |-| + 4*cos |-|
\2/ \2/
$$- 4 \cos^{4}{\left(\frac{t}{2} \right)} + 4 \cos^{2}{\left(\frac{t}{2} \right)}$$
2 2 1 sin (t) cos (t) - + ------- - ------- 2 2 2
$$\frac{\sin^{2}{\left(t \right)}}{2} - \frac{\cos^{2}{\left(t \right)}}{2} + \frac{1}{2}$$
2/t\
4*tan |-|
\2/
--------------
2
/ 2/t\\
|1 + tan |-||
\ \2//
$$\frac{4 \tan^{2}{\left(\frac{t}{2} \right)}}{\left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}}$$
/ 0 for t mod pi = 0 | < 2 |sin (t) otherwise \
$$\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\sin^{2}{\left(t \right)} & \text{otherwise} \end{cases}$$
// 1 for t mod 2*pi = 0\
|| |
1 - |< 2 |
||cos (t) otherwise |
\\ /
$$\left(- \begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\cos^{2}{\left(t \right)} & \text{otherwise} \end{cases}\right) + 1$$
2
/ 2/t\\
|-1 + cot |-||
\ \2//
1 - ---------------
2
/ 2/t\\
|1 + cot |-||
\ \2//
$$- \frac{\left(\cot^{2}{\left(\frac{t}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} + 1$$
2
/ 2/t\\
|1 - tan |-||
\ \2//
1 - --------------
2
/ 2/t\\
|1 + tan |-||
\ \2//
$$- \frac{\left(- \tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} + 1$$
// 1 for t mod 2*pi = 0\
|| |
|| 1 |
1 - |<------- otherwise |
|| 2 |
||sec (t) |
\\ /
$$\left(- \begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{1}{\sec^{2}{\left(t \right)}} & \text{otherwise} \end{cases}\right) + 1$$
2/t pi\
4*tan |- + --|
\2 4 /
1 - -------------------
2
/ 2/t pi\\
|1 + tan |- + --||
\ \2 4 //
$$1 - \frac{4 \tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
// 1 for t mod 2*pi = 0\
|| |
1 - |< 2/ pi\ |
||sin |t + --| otherwise |
\\ \ 2 / /
$$\left(- \begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\sin^{2}{\left(t + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + 1$$
2
/ 1 \
|1 - -------|
| 2/t\|
| cot |-||
\ \2//
1 - --------------
2
/ 1 \
|1 + -------|
| 2/t\|
| cot |-||
\ \2//
$$- \frac{\left(1 - \frac{1}{\cot^{2}{\left(\frac{t}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{t}{2} \right)}}\right)^{2}} + 1$$
// 1 for t mod 2*pi = 0\
|| |
|| 1 |
1 - |<------------ otherwise |
|| 2/pi \ |
||csc |-- - t| |
\\ \2 / /
$$\left(- \begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{1}{\csc^{2}{\left(- t + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1$$
2 4/t\
16*sin (t)*sin |-|
\2/
----------------------
2
/ 2 4/t\\
|sin (t) + 4*sin |-||
\ \2//
$$\frac{16 \sin^{4}{\left(\frac{t}{2} \right)} \sin^{2}{\left(t \right)}}{\left(4 \sin^{4}{\left(\frac{t}{2} \right)} + \sin^{2}{\left(t \right)}\right)^{2}}$$
/ 0 for t mod pi = 0 | | 2/t\ | 4*cot |-| | \2/ <-------------- otherwise | 2 |/ 2/t\\ ||1 + cot |-|| |\ \2// \
$$\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{t}{2} \right)}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}$$
2
/ 4/t\\
| 4*sin |-||
| \2/|
|1 - ---------|
| 2 |
\ sin (t) /
1 - ----------------
2
/ 4/t\\
| 4*sin |-||
| \2/|
|1 + ---------|
| 2 |
\ sin (t) /
$$- \frac{\left(- \frac{4 \sin^{4}{\left(\frac{t}{2} \right)}}{\sin^{2}{\left(t \right)}} + 1\right)^{2}}{\left(\frac{4 \sin^{4}{\left(\frac{t}{2} \right)}}{\sin^{2}{\left(t \right)}} + 1\right)^{2}} + 1$$
// 1 for t mod 2*pi = 0\
|| |
||/ 1 for t mod 2*pi = 0 |
1 - |<| |
||< 2 otherwise |
|||cos (t) otherwise |
\\\ /
$$\left(- \begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\cos^{2}{\left(t \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + 1$$
// 1 for t mod 2*pi = 0\
|| |
|| 2 |
||/ 2/t\\ |
|||-1 + cot |-|| |
1 - |<\ \2// |
||--------------- otherwise |
|| 2 |
|| / 2/t\\ |
|| |1 + cot |-|| |
\\ \ \2// /
$$\left(- \begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{t}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) + 1$$
// / pi\ \
|| 0 for |t + --| mod pi = 0|
|| \ 2 / |
1 - |< |
|| 2 2/t pi\ |
||(1 + sin(t)) *cot |- + --| otherwise |
\\ \2 4 / /
$$\left(- \begin{cases} 0 & \text{for}\: \left(t + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(t \right)} + 1\right)^{2} \cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right) + 1$$
// 1 for t mod 2*pi = 0\
|| |
|| 2 |
||/ 2/t\\ |
|||1 - tan |-|| |
1 - |<\ \2// |
||-------------- otherwise |
|| 2 |
||/ 2/t\\ |
|||1 + tan |-|| |
\\\ \2// /
$$\left(- \begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(- \tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) + 1$$
// 1 for t mod 2*pi = 0\
|| |
|| 2 |
||/ 1 \ |
|||-1 + -------| |
||| 2/t\| |
||| tan |-|| |
1 - |<\ \2// |
||--------------- otherwise |
|| 2 |
|| / 1 \ |
|| |1 + -------| |
|| | 2/t\| |
|| | tan |-|| |
\\ \ \2// /
$$\left(- \begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{t}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{t}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right) + 1$$
2
/ 2/t pi\\
| cos |- - --||
| \2 2 /|
|1 - ------------|
| 2/t\ |
| cos |-| |
\ \2/ /
1 - -------------------
2
/ 2/t pi\\
| cos |- - --||
| \2 2 /|
|1 + ------------|
| 2/t\ |
| cos |-| |
\ \2/ /
$$- \frac{\left(1 - \frac{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} \right)}}\right)^{2}}{\left(1 + \frac{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} \right)}}\right)^{2}} + 1$$
2
/ 2/t\ \
| sec |-| |
| \2/ |
|1 - ------------|
| 2/t pi\|
| sec |- - --||
\ \2 2 //
1 - -------------------
2
/ 2/t\ \
| sec |-| |
| \2/ |
|1 + ------------|
| 2/t pi\|
| sec |- - --||
\ \2 2 //
$$- \frac{\left(- \frac{\sec^{2}{\left(\frac{t}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}}{\left(\frac{\sec^{2}{\left(\frac{t}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}} + 1$$
2
/ 2/pi t\\
| csc |-- - -||
| \2 2/|
|1 - ------------|
| 2/t\ |
| csc |-| |
\ \2/ /
1 - -------------------
2
/ 2/pi t\\
| csc |-- - -||
| \2 2/|
|1 + ------------|
| 2/t\ |
| csc |-| |
\ \2/ /
$$- \frac{\left(1 - \frac{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{t}{2} \right)}}\right)^{2}}{\left(1 + \frac{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{t}{2} \right)}}\right)^{2}} + 1$$
// / pi\ \
|| 0 for |t + --| mod pi = 0|
|| \ 2 / |
|| |
|| 2/t pi\ |
|| 4*cot |- + --| |
1 - |< \2 4 / |
||------------------- otherwise |
|| 2 |
||/ 2/t pi\\ |
|||1 + cot |- + --|| |
||\ \2 4 // |
\\ /
$$\left(- \begin{cases} 0 & \text{for}\: \left(t + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)}}{\left(\cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) + 1$$
// 1 for t mod 2*pi = 0\
|| |
|| 2 |
||/ 2 4/t\\ |
|||sin (t) - 4*sin |-|| |
1 - |<\ \2// |
||---------------------- otherwise |
|| 2 |
||/ 2 4/t\\ |
|||sin (t) + 4*sin |-|| |
\\\ \2// /
$$\left(- \begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(- 4 \sin^{4}{\left(\frac{t}{2} \right)} + \sin^{2}{\left(t \right)}\right)^{2}}{\left(4 \sin^{4}{\left(\frac{t}{2} \right)} + \sin^{2}{\left(t \right)}\right)^{2}} & \text{otherwise} \end{cases}\right) + 1$$
// 1 for t mod 2*pi = 0\
|| |
|| 2 |
||/ 2 \ |
||| sin (t) | |
|||-1 + ---------| |
||| 4/t\| |
||| 4*sin |-|| |
1 - |<\ \2// |
||----------------- otherwise |
|| 2 |
|| / 2 \ |
|| | sin (t) | |
|| |1 + ---------| |
|| | 4/t\| |
|| | 4*sin |-|| |
\\ \ \2// /
$$\left(- \begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sin^{2}{\left(t \right)}}{4 \sin^{4}{\left(\frac{t}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sin^{2}{\left(t \right)}}{4 \sin^{4}{\left(\frac{t}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right) + 1$$
// 1 for t mod 2*pi = 0\
|| |
||/ 1 for t mod 2*pi = 0 |
||| |
||| 2 |
|||/ 2/t\\ |
1 - |<||-1 + cot |-|| |
||<\ \2// otherwise |
|||--------------- otherwise |
||| 2 |
||| / 2/t\\ |
||| |1 + cot |-|| |
\\\ \ \2// /
$$\left(- \begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{t}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + 1$$
// 1 for t mod 2*pi = 0\
|| |
|| 2 |
||/ 2/t\ \ |
||| cos |-| | |
||| \2/ | |
|||-1 + ------------| |
||| 2/t pi\| |
||| cos |- - --|| |
1 - |<\ \2 2 // |
||-------------------- otherwise |
|| 2 |
||/ 2/t\ \ |
||| cos |-| | |
||| \2/ | |
|||1 + ------------| |
||| 2/t pi\| |
||| cos |- - --|| |
\\\ \2 2 // /
$$\left(- \begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(\frac{\cos^{2}{\left(\frac{t}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\cos^{2}{\left(\frac{t}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) + 1$$
// 1 for t mod 2*pi = 0\
|| |
|| 2 |
||/ 2/t pi\\ |
||| sec |- - --|| |
||| \2 2 /| |
|||-1 + ------------| |
||| 2/t\ | |
||| sec |-| | |
1 - |<\ \2/ / |
||-------------------- otherwise |
|| 2 |
||/ 2/t pi\\ |
||| sec |- - --|| |
||| \2 2 /| |
|||1 + ------------| |
||| 2/t\ | |
||| sec |-| | |
\\\ \2/ / /
$$\left(- \begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right) + 1$$
// 1 for t mod 2*pi = 0\
|| |
|| 2 |
||/ 2/t\ \ |
||| csc |-| | |
||| \2/ | |
|||-1 + ------------| |
||| 2/pi t\| |
||| csc |-- - -|| |
1 - |<\ \2 2// |
||-------------------- otherwise |
|| 2 |
||/ 2/t\ \ |
||| csc |-| | |
||| \2/ | |
|||1 + ------------| |
||| 2/pi t\| |
||| csc |-- - -|| |
\\\ \2 2// /
$$\left(- \begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{t}{2} \right)}}{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\csc^{2}{\left(\frac{t}{2} \right)}}{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) + 1$$
1 - Piecewise((1, Mod(t = 2*pi, 0)), ((-1 + csc(t/2)^2/csc(pi/2 - t/2)^2)^2/(1 + csc(t/2)^2/csc(pi/2 - t/2)^2)^2, True))