Господин Экзамен
Общий знаменатель tan(pi/2-a)^(2)-1-sin(2*a)/1+sin(2*a)
Решение
Вы ввели
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2/pi \ sin(2*a)
tan |-- - a| - 1 - -------- + sin(2*a)
\2 / 1
$$\tan^{2}{\left(- a + \frac{\pi}{2} \right)} - \frac{\sin{\left(2 a \right)}}{1} + \sin{\left(2 a \right)} - 1$$
tan(pi/2 - a)^2 - 1*1 - sin(2*a)/1 + sin(2*a)
Объединение рациональных выражений
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2/pi - 2*a\
-1 + tan |--------|
\ 2 /
$$\tan^{2}{\left(\frac{- 2 a + \pi}{2} \right)} - 1$$
-1 + tan((pi - 2*a)/2)^2
Комбинаторика
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(1 + cot(a))*(-1 + cot(a))
$$\left(\cot{\left(a \right)} - 1\right) \left(\cot{\left(a \right)} + 1\right)$$
(1 + cot(a))*(-1 + cot(a))
Степени
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2 -1 + cot (a)
$$\cot^{2}{\left(a \right)} - 1$$
2/pi \
-1 + tan |-- - a|
\2 /
$$\tan^{2}{\left(- a + \frac{\pi}{2} \right)} - 1$$
2
/ /pi \ / pi\\
| I*|-- - a| I*|a - --||
| \2 / \ 2 /|
\- e + e /
-1 - ------------------------------
2
/ / pi\ /pi \\
| I*|a - --| I*|-- - a||
| \ 2 / \2 /|
\e + e /
$$- \frac{\left(- e^{i \left(- a + \frac{\pi}{2}\right)} + e^{i \left(a - \frac{\pi}{2}\right)}\right)^{2}}{\left(e^{i \left(- a + \frac{\pi}{2}\right)} + e^{i \left(a - \frac{\pi}{2}\right)}\right)^{2}} - 1$$
-1 - (-exp(i*(pi/2 - a)) + exp(i*(a - pi/2)))^2/(exp(i*(a - pi/2)) + exp(i*(pi/2 - a)))^2
Собрать выражение
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2 -1 + cot (a)
$$\cot^{2}{\left(a \right)} - 1$$
2/pi \
-1 + tan |-- - a|
\2 /
$$\tan^{2}{\left(- a + \frac{\pi}{2} \right)} - 1$$
-1 + tan(pi/2 - a)^2
Тригонометрическая часть
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2 -1 + cot (a)
$$\cot^{2}{\left(a \right)} - 1$$
1
-1 + -------
2
tan (a)
$$-1 + \frac{1}{\tan^{2}{\left(a \right)}}$$
2
csc (a)
-1 + -------
2
sec (a)
$$\frac{\csc^{2}{\left(a \right)}}{\sec^{2}{\left(a \right)}} - 1$$
2*cos(2*a) ------------ 1 - cos(2*a)
$$\frac{2 \cos{\left(2 a \right)}}{- \cos{\left(2 a \right)} + 1}$$
2
cos (a)
-1 + -------
2
sin (a)
$$-1 + \frac{\cos^{2}{\left(a \right)}}{\sin^{2}{\left(a \right)}}$$
4
csc (a)
-1 + -----------
2
4*csc (2*a)
$$\frac{\csc^{4}{\left(a \right)}}{4 \csc^{2}{\left(2 a \right)}} - 1$$
2
sin (2*a)
-1 + ---------
4
4*sin (a)
$$-1 + \frac{\sin^{2}{\left(2 a \right)}}{4 \sin^{4}{\left(a \right)}}$$
2
cos (a)
-1 + ------------
2/ pi\
cos |a - --|
\ 2 /
$$\frac{\cos^{2}{\left(a \right)}}{\cos^{2}{\left(a - \frac{\pi}{2} \right)}} - 1$$
2/ pi\
sec |a - --|
\ 2 /
-1 + ------------
2
sec (a)
$$-1 + \frac{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(a \right)}}$$
2/ pi\
sin |a + --|
\ 2 /
-1 + ------------
2
sin (a)
$$-1 + \frac{\sin^{2}{\left(a + \frac{\pi}{2} \right)}}{\sin^{2}{\left(a \right)}}$$
2
csc (a)
-1 + ------------
2/pi \
csc |-- - a|
\2 /
$$\frac{\csc^{2}{\left(a \right)}}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} - 1$$
2/pi \
sec |-- - a|
\2 /
-1 + ------------
2
sec (a)
$$-1 + \frac{\sec^{2}{\left(- a + \frac{\pi}{2} \right)}}{\sec^{2}{\left(a \right)}}$$
2
csc (pi - a)
-1 + ------------
2/pi \
csc |-- - a|
\2 /
$$\frac{\csc^{2}{\left(- a + \pi \right)}}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} - 1$$
4/ pi\
sec |a - --|
\ 2 /
-1 + ----------------
2/ pi\
4*sec |2*a - --|
\ 2 /
$$\frac{\sec^{4}{\left(a - \frac{\pi}{2} \right)}}{4 \sec^{2}{\left(2 a - \frac{\pi}{2} \right)}} - 1$$
2/ pi\
cos |2*a - --|
\ 2 /
-1 + --------------
4/ pi\
4*cos |a - --|
\ 2 /
$$-1 + \frac{\cos^{2}{\left(2 a - \frac{\pi}{2} \right)}}{4 \cos^{4}{\left(a - \frac{\pi}{2} \right)}}$$
1 1
-2 + --------- + ---------
2/a\ 2/a\
4*cos |-| 4*sin |-|
\2/ \2/
$$-2 + \frac{1}{4 \cos^{2}{\left(\frac{a}{2} \right)}} + \frac{1}{4 \sin^{2}{\left(\frac{a}{2} \right)}}$$
2
/ 2/a\\
|1 - tan |-||
\ \2//
-1 + --------------
2/a\
4*tan |-|
\2/
$$\frac{\left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}{4 \tan^{2}{\left(\frac{a}{2} \right)}} - 1$$
2/a\
tan |-|
3 1 \2/
- - + --------- + -------
2 2/a\ 4
4*tan |-|
\2/
$$\frac{\tan^{2}{\left(\frac{a}{2} \right)}}{4} - \frac{3}{2} + \frac{1}{4 \tan^{2}{\left(\frac{a}{2} \right)}}$$
4
/ 2/a\\ 2
|1 + tan |-|| *tan (a)
\ \2//
-1 + -------------------------
2
/ 2 \ 4/a\
16*\1 + tan (a)/ *tan |-|
\2/
$$\frac{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4} \tan^{2}{\left(a \right)}}{16 \left(\tan^{2}{\left(a \right)} + 1\right)^{2} \tan^{4}{\left(\frac{a}{2} \right)}} - 1$$
2
/ 2/a\\ 4/a\
4*|1 - tan |-|| *cos |-|
\ \2// \2/
-1 + ---------------------------------
2
/ 2/a pi\\ 2
|1 - cot |- + --|| *(1 + sin(a))
\ \2 4 //
$$\frac{4 \left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2} \cos^{4}{\left(\frac{a}{2} \right)}}{\left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\sin{\left(a \right)} + 1\right)^{2}} - 1$$
2
/ 2/a\\ 2/a pi\
|1 + cot |-|| *tan |- + --|
\ \2// \2 4 /
-1 + ---------------------------
2
/ 2/a pi\\ 2/a\
|1 + tan |- + --|| *cot |-|
\ \2 4 // \2/
$$-1 + \frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2} \tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \cot^{2}{\left(\frac{a}{2} \right)}}$$
2 2
/ 2/a pi\\ / 2/a\\
|1 + tan |- + --|| *|-1 + cot |-||
\ \2 4 // \ \2//
-1 + -----------------------------------
2 2
/ 2/a\\ / 2/a pi\\
|1 + cot |-|| *|-1 + tan |- + --||
\ \2// \ \2 4 //
$$-1 + \frac{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1\right)^{2} \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}$$
2 2
/ 2/a pi\\ / 2/a\\
|1 + cot |- + --|| *|1 - tan |-||
\ \2 4 // \ \2//
-1 + ----------------------------------
2 2
/ 2/a\\ / 2/a pi\\
|1 + tan |-|| *|1 - cot |- + --||
\ \2// \ \2 4 //
$$\frac{\left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2} \left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}{\left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} - 1$$
// zoo for a mod pi = 0\
|| |
// 0 for 2*a mod pi = 0\ || 4/a\ |
|| | || tan |-| |
|<1 - cos(4*a) |*|< \2/ |
||------------ otherwise | ||---------- otherwise |
\\ 2 / || 8/a\ |
||16*sin |-| |
\\ \2/ /
-1 + -------------------------------------------------------------------
4
$$\left(\frac{\left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{- \cos{\left(4 a \right)} + 1}{2} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{\tan^{4}{\left(\frac{a}{2} \right)}}{16 \sin^{8}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)}{4}\right) - 1$$
// / 3*pi\ \
|| 1 for |a + ----| mod 2*pi = 0|
|| \ 2 / |
|| |
// 1 for a mod 2*pi = 0\ || 1 |
|| | ||-1 + ------- |
-1 + |< 2 |*|< 2/a\ |
||cos (a) otherwise | || sin |-| |
\\ / || \2/ |
||------------ otherwise |
|| 4/a\ |
|| 4*cos |-| |
\\ \2/ /
$$\left(\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\sin^{2}{\left(\frac{a}{2} \right)}}}{4 \cos^{4}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - 1$$
// zoo for a mod pi = 0\
// 0 for 2*a mod pi = 0\ || |
|| | || 4 |
|| 2 | ||/ 2/a\\ |
|| 4*cot (a) | |||1 + cot |-|| |
|<-------------- otherwise |*|<\ \2// |
|| 2 | ||-------------- otherwise |
||/ 2 \ | || 4/a\ |
||\1 + cot (a)/ | || 16*cot |-| |
\\ / || \2/ |
\\ /
-1 + -------------------------------------------------------------------------
4
$$\left(\frac{\left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(a \right)}}{\left(\cot^{2}{\left(a \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4}}{16 \cot^{4}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)}{4}\right) - 1$$
// / 3*pi\ \
// 1 for a mod 2*pi = 0\ || 1 for |a + ----| mod 2*pi = 0|
|| | || \ 2 / |
|| 2 | || |
||/ 2/a\\ | || 2 |
|||-1 + cot |-|| | ||/ 2/a pi\\ |
-1 + |<\ \2// |*|<|1 + tan |- + --|| |
||--------------- otherwise | ||\ \2 4 // |
|| 2 | ||-------------------- otherwise |
|| / 2/a\\ | || 2 |
|| |1 + cot |-|| | ||/ 2/a pi\\ |
\\ \ \2// / |||-1 + tan |- + --|| |
\\\ \2 4 // /
$$\left(\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) - 1$$
-1 + Piecewise((1, Mod(a = 2*pi, 0)), ((-1 + cot(a/2)^2)^2/(1 + cot(a/2)^2)^2, True))*Piecewise((1, Mod(a + 3*pi/2 = 2*pi, 0)), ((1 + tan(a/2 + pi/4)^2)^2/(-1 + tan(a/2 + pi/4)^2)^2, True))