Господин Экзамен
tan(a)^2*cot(a)^2-sin(a)^2 если a=2
Решение
Вы ввели
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2 2 2 tan (a)*cot (a) - sin (a)
$$\tan^{2}{\left(a \right)} \cot^{2}{\left(a \right)} - \sin^{2}{\left(a \right)}$$
tan(a)^2*cot(a)^2 - sin(a)^2
Подстановка условия
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tan(a)^2*cot(a)^2 - sin(a)^2 при a = 2
подставляем
2 2 2 tan (a)*cot (a) - sin (a)
$$\tan^{2}{\left(a \right)} \cot^{2}{\left(a \right)} - \sin^{2}{\left(a \right)}$$
2 cos (a)
$$\cos^{2}{\left(a \right)}$$
переменные
a = 2
$$a = 2$$
2 cos ((2))
$$\cos^{2}{\left((2) \right)}$$
2 cos (2)
$$\cos^{2}{\left(2 \right)}$$
cos(2)^2
Степени
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2 2
/ -I*a I*a\ / I*a -I*a\ 2
\- e + e / \- e + e / *cot (a)
----------------- - -------------------------
4 2
/ I*a -I*a\
\e + e /
$$- \frac{\left(- e^{i a} + e^{- i a}\right)^{2} \cot^{2}{\left(a \right)}}{\left(e^{i a} + e^{- i a}\right)^{2}} + \frac{\left(e^{i a} - e^{- i a}\right)^{2}}{4}$$
(-exp(-i*a) + exp(i*a))^2/4 - (-exp(i*a) + exp(-i*a))^2*cot(a)^2/(exp(i*a) + exp(-i*a))^2
Комбинаторика
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(-sin(a) + cot(a)*tan(a))*(cot(a)*tan(a) + sin(a))
$$\left(\tan{\left(a \right)} \cot{\left(a \right)} - \sin{\left(a \right)}\right) \left(\tan{\left(a \right)} \cot{\left(a \right)} + \sin{\left(a \right)}\right)$$
(-sin(a) + cot(a)*tan(a))*(cot(a)*tan(a) + sin(a))
Собрать выражение
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1 cos(2*a) - + -------- 2 2
$$\frac{\cos{\left(2 a \right)}}{2} + \frac{1}{2}$$
1/2 + cos(2*a)/2
Тригонометрическая часть
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2 cos (a)
$$\cos^{2}{\left(a \right)}$$
2 1 - sin (a)
$$- \sin^{2}{\left(a \right)} + 1$$
1 ------- 2 sec (a)
$$\frac{1}{\sec^{2}{\left(a \right)}}$$
1
1 - -------
2
csc (a)
$$1 - \frac{1}{\csc^{2}{\left(a \right)}}$$
2/ pi\
sin |a + --|
\ 2 /
$$\sin^{2}{\left(a + \frac{\pi}{2} \right)}$$
1
1 - ------------
2
csc (pi - a)
$$1 - \frac{1}{\csc^{2}{\left(- a + \pi \right)}}$$
2/ pi\
1 - cos |a - --|
\ 2 /
$$- \cos^{2}{\left(a - \frac{\pi}{2} \right)} + 1$$
1 cos(2*a) - + -------- 2 2
$$\frac{\cos{\left(2 a \right)}}{2} + \frac{1}{2}$$
1
------------
2/pi \
csc |-- - a|
\2 /
$$\frac{1}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}}$$
1
1 - ------------
2/ pi\
sec |a - --|
\ 2 /
$$1 - \frac{1}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}$$
1
1 - ------------
2/pi \
sec |-- - a|
\2 /
$$1 - \frac{1}{\sec^{2}{\left(- a + \frac{\pi}{2} \right)}}$$
1 cos(2*a) 2 2 - - + -------- + cot (a)*tan (a) 2 2
$$\tan^{2}{\left(a \right)} \cot^{2}{\left(a \right)} + \frac{\cos{\left(2 a \right)}}{2} - \frac{1}{2}$$
2/a\
4*tan |-|
\2/
1 - --------------
2
/ 2/a\\
|1 + tan |-||
\ \2//
$$1 - \frac{4 \tan^{2}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}$$
2/a\
4*cot |-|
\2/
1 - --------------
2
/ 2/a\\
|1 + cot |-||
\ \2//
$$1 - \frac{4 \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}$$
/ 1 for a mod 2*pi = 0 | < 2 |cos (a) otherwise \
$$\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos^{2}{\left(a \right)} & \text{otherwise} \end{cases}$$
4
1 - ----------------------
2
/ 1 \ 2/a\
|1 + -------| *cot |-|
| 2/a\| \2/
| cot |-||
\ \2//
$$1 - \frac{4}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right)^{2} \cot^{2}{\left(\frac{a}{2} \right)}}$$
2
/ 2/a\\
|1 - tan |-||
\ \2//
--------------
2
/ 2/a\\
|1 + tan |-||
\ \2//
$$\frac{\left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}$$
// 0 for a mod pi = 0\
|| |
1 - |< 2 |
||sin (a) otherwise |
\\ /
$$\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1$$
2
/ 2/a pi\\ 2
|1 - cot |- + --|| *(1 + sin(a))
\ \2 4 //
1 - ---------------------------------
4
$$- \frac{\left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\sin{\left(a \right)} + 1\right)^{2}}{4} + 1$$
// 0 for a mod pi = 0\
|| |
|| 1 |
1 - |<------- otherwise |
|| 2 |
||csc (a) |
\\ /
$$\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\csc^{2}{\left(a \right)}} & \text{otherwise} \end{cases}\right) + 1$$
// 0 for a mod pi = 0\
|| |
1 - |< 2/ pi\ |
||cos |a - --| otherwise |
\\ \ 2 / /
$$\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\cos^{2}{\left(a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + 1$$
2 2 1 cos (a) sin (a) 2 2 - - + ------- - ------- + cot (a)*tan (a) 2 2 2
$$\tan^{2}{\left(a \right)} \cot^{2}{\left(a \right)} - \frac{\sin^{2}{\left(a \right)}}{2} + \frac{\cos^{2}{\left(a \right)}}{2} - \frac{1}{2}$$
// 0 for a mod pi = 0\
|| |
|| 1 |
1 - |<------------ otherwise |
|| 2/ pi\ |
||sec |a - --| |
\\ \ 2 / /
$$\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1$$
2
/ 2/a pi\\
|-1 + tan |- + --||
\ \2 4 //
1 - --------------------
2
/ 2/a pi\\
|1 + tan |- + --||
\ \2 4 //
$$- \frac{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} + 1$$
2
/ 2/a pi\\
|1 - cot |- + --||
\ \2 4 //
1 - -------------------
2
/ 2/a pi\\
|1 + cot |- + --||
\ \2 4 //
$$- \frac{\left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} + 1$$
2 4/a\
16*sin (a)*sin |-|
\2/
1 - ----------------------
2
/ 2 4/a\\
|sin (a) + 4*sin |-||
\ \2//
$$- \frac{16 \sin^{4}{\left(\frac{a}{2} \right)} \sin^{2}{\left(a \right)}}{\left(4 \sin^{4}{\left(\frac{a}{2} \right)} + \sin^{2}{\left(a \right)}\right)^{2}} + 1$$
4/a\
16*sin |-|
\2/
1 - ------------------------
2
/ 4/a\\
| 4*sin |-||
| \2/| 2
|1 + ---------| *sin (a)
| 2 |
\ sin (a) /
$$1 - \frac{16 \sin^{4}{\left(\frac{a}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right)^{2} \sin^{2}{\left(a \right)}}$$
// 0 for a mod pi = 0\
|| |
|| 2/a\ |
|| 4*cot |-| |
|| \2/ |
1 - |<-------------- otherwise |
|| 2 |
||/ 2/a\\ |
|||1 + cot |-|| |
||\ \2// |
\\ /
$$\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) + 1$$
// 0 for a mod pi = 0\
|| |
|| 2/a\ |
|| 4*tan |-| |
|| \2/ |
1 - |<-------------- otherwise |
|| 2 |
||/ 2/a\\ |
|||1 + tan |-|| |
||\ \2// |
\\ /
$$\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{4 \tan^{2}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) + 1$$
/ 1 for a mod 2*pi = 0 | | 2 |/ 2/a\\ ||-1 + cot |-|| <\ \2// |--------------- otherwise | 2 | / 2/a\\ | |1 + cot |-|| \ \ \2//
$$\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}$$
// 0 for a mod pi = 0\
|| |
||/ 0 for a mod pi = 0 |
1 - |<| |
||< 2 otherwise |
|||sin (a) otherwise |
\\\ /
$$\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin^{2}{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + 1$$
// / 3*pi\ \
|| 1 for |a + ----| mod 2*pi = 0|
|| \ 2 / |
1 - |< |
|| 4/a\ 2/a\ |
||- 4*cos |-| + 4*cos |-| otherwise |
\\ \2/ \2/ /
$$\left(- \begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\- 4 \cos^{4}{\left(\frac{a}{2} \right)} + 4 \cos^{2}{\left(\frac{a}{2} \right)} & \text{otherwise} \end{cases}\right) + 1$$
// 0 for a mod pi = 0\
|| |
|| 4 |
||---------------------- otherwise |
|| 2 |
1 - |
$$\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{4}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right)^{2} \tan^{2}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1$$
2/a pi\
4*cos |- - --|
\2 2 /
1 - ---------------------------
2
/ 2/a pi\\
| cos |- - --||
| \2 2 /| 2/a\
|1 + ------------| *cos |-|
| 2/a\ | \2/
| cos |-| |
\ \2/ /
$$1 - \frac{4 \cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right)^{2} \cos^{2}{\left(\frac{a}{2} \right)}}$$
2/a\
4*sec |-|
\2/
1 - --------------------------------
2
/ 2/a\ \
| sec |-| |
| \2/ | 2/a pi\
|1 + ------------| *sec |- - --|
| 2/a pi\| \2 2 /
| sec |- - --||
\ \2 2 //
$$1 - \frac{4 \sec^{2}{\left(\frac{a}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}$$
2/pi a\
4*csc |-- - -|
\2 2/
1 - ---------------------------
2
/ 2/pi a\\
| csc |-- - -||
| \2 2/| 2/a\
|1 + ------------| *csc |-|
| 2/a\ | \2/
| csc |-| |
\ \2/ /
$$1 - \frac{4 \csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right)^{2} \csc^{2}{\left(\frac{a}{2} \right)}}$$
// 0 for a mod pi = 0\
|| |
|| 2 |
|| sin (a) |
||------------------------ otherwise |
|| 2 |
1 - |
$$\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{\sin^{2}{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1$$
// 0 for a mod pi = 0\
|| |
||/ 0 for a mod pi = 0 |
||| |
||| 2/a\ |
||| 4*cot |-| |
1 - |<| \2/ |
||<-------------- otherwise otherwise |
||| 2 |
|||/ 2/a\\ |
||||1 + cot |-|| |
|||\ \2// |
\\\ /
$$\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + 1$$
// / 3*pi\ \
|| 1 for |a + ----| mod 2*pi = 0|
|| \ 2 / |
|| |
|| 2 |
||/ 2/a pi\\ |
1 - |<|-1 + tan |- + --|| |
||\ \2 4 // |
||-------------------- otherwise |
|| 2 |
||/ 2/a pi\\ |
|||1 + tan |- + --|| |
\\\ \2 4 // /
$$\left(- \begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) + 1$$
// 0 for a mod pi = 0\
|| |
|| 2/a pi\ |
|| 4*sec |- - --| |
|| \2 2 / |
||--------------------------- otherwise |
|| 2 |
1 - |
$$\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{4 \sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right)^{2} \sec^{2}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1$$
// 0 for a mod pi = 0\
|| |
|| 2/a\ |
|| 4*cos |-| |
|| \2/ |
||-------------------------------- otherwise |
|| 2 |
1 - |
$$\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{4 \cos^{2}{\left(\frac{a}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1$$
// 0 for a mod pi = 0\
|| |
|| 2/a\ |
|| 4*csc |-| |
|| \2/ |
||-------------------------------- otherwise |
|| 2 |
1 - |
$$\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{4 \csc^{2}{\left(\frac{a}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2} \csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1$$
1 - Piecewise((0, Mod(a = pi, 0)), (4*csc(a/2)^2/((1 + csc(a/2)^2/csc(pi/2 - a/2)^2)^2*csc(pi/2 - a/2)^2), True))