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