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