Господин Экзамен
2*sin(a)-2*cos(a) если a=3
Решение
Вы ввели
[src]
2*sin(a) - 2*cos(a)
$$2 \sin{\left(a \right)} - 2 \cos{\left(a \right)}$$
2*sin(a) - 2*cos(a)
Общее упрощение
[src]
___ / pi\
-2*\/ 2 *cos|a + --|
\ 4 /
$$- 2 \sqrt{2} \cos{\left(a + \frac{\pi}{4} \right)}$$
-2*sqrt(2)*cos(a + pi/4)
Подстановка условия
[src]
2*sin(a) - 2*cos(a) при a = 3
подставляем
2*sin(a) - 2*cos(a)
$$2 \sin{\left(a \right)} - 2 \cos{\left(a \right)}$$
___ / pi\
-2*\/ 2 *cos|a + --|
\ 4 /
$$- 2 \sqrt{2} \cos{\left(a + \frac{\pi}{4} \right)}$$
переменные
a = 3
$$a = 3$$
___ / pi\
-2*\/ 2 *cos|(3) + --|
\ 4 /
$$- 2 \sqrt{2} \cos{\left((3) + \frac{\pi}{4} \right)}$$
___ / pi\
-2*\/ 2 *cos|3 + --|
\ 4 /
$$- 2 \sqrt{2} \cos{\left(\frac{\pi}{4} + 3 \right)}$$
-2*sqrt(2)*cos(3 + pi/4)
Степени
[src]
I*a -I*a / -I*a I*a\ - e - e - I*\- e + e /
$$- i \left(e^{i a} - e^{- i a}\right) - e^{i a} - e^{- i a}$$
-exp(i*a) - exp(-i*a) - i*(-exp(-i*a) + exp(i*a))
Тригонометрическая часть
[src]
___ / pi\
-2*\/ 2 *cos|a + --|
\ 4 /
$$- 2 \sqrt{2} \cos{\left(a + \frac{\pi}{4} \right)}$$
___ / 3*pi\
-2*\/ 2 *sin|a + ----|
\ 4 /
$$- 2 \sqrt{2} \sin{\left(a + \frac{3 \pi}{4} \right)}$$
___ -2*\/ 2 ----------- / pi\ sec|a + --| \ 4 /
$$- \frac{2 \sqrt{2}}{\sec{\left(a + \frac{\pi}{4} \right)}}$$
___ -2*\/ 2 ------------ / pi\ csc|-a + --| \ 4 /
$$- \frac{2 \sqrt{2}}{\csc{\left(- a + \frac{\pi}{4} \right)}}$$
/ pi\
- 2*sin|a + --| + 2*sin(a)
\ 2 /
$$2 \sin{\left(a \right)} - 2 \sin{\left(a + \frac{\pi}{2} \right)}$$
/ pi\
-2*cos(a) + 2*cos|a - --|
\ 2 /
$$- 2 \cos{\left(a \right)} + 2 \cos{\left(a - \frac{\pi}{2} \right)}$$
2 2 - ------ + ------ sec(a) csc(a)
$$- \frac{2}{\sec{\left(a \right)}} + \frac{2}{\csc{\left(a \right)}}$$
2/a\
2 - 4*cos |-| + 2*sin(a)
\2/
$$- 4 \cos^{2}{\left(\frac{a}{2} \right)} + 2 \sin{\left(a \right)} + 2$$
2 2
- ------ + -----------
sec(a) / pi\
sec|a - --|
\ 2 /
$$\frac{2}{\sec{\left(a - \frac{\pi}{2} \right)}} - \frac{2}{\sec{\left(a \right)}}$$
2 2
- ------ + -----------
sec(a) /pi \
sec|-- - a|
\2 /
$$\frac{2}{\sec{\left(- a + \frac{\pi}{2} \right)}} - \frac{2}{\sec{\left(a \right)}}$$
2 2
- ----------- + ------
/pi \ csc(a)
csc|-- - a|
\2 /
$$- \frac{2}{\csc{\left(- a + \frac{\pi}{2} \right)}} + \frac{2}{\csc{\left(a \right)}}$$
2 2
- ----------- + -----------
/pi \ csc(pi - a)
csc|-- - a|
\2 /
$$- \frac{2}{\csc{\left(- a + \frac{\pi}{2} \right)}} + \frac{2}{\csc{\left(- a + \pi \right)}}$$
/a\
-2*cos(a) + 2*(1 + cos(a))*tan|-|
\2/
$$2 \left(\cos{\left(a \right)} + 1\right) \tan{\left(\frac{a}{2} \right)} - 2 \cos{\left(a \right)}$$
/ 2/a pi\\
-2*cos(a) + |1 - cot |- + --||*(1 + sin(a))
\ \2 4 //
$$\left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(a \right)} + 1\right) - 2 \cos{\left(a \right)}$$
___ / 2/a pi\\
-2*\/ 2 *|1 - tan |- + --||
\ \2 8 //
---------------------------
2/a pi\
1 + tan |- + --|
\2 8 /
$$- \frac{2 \sqrt{2} \cdot \left(- \tan^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} + 1\right)}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} + 1}$$
/ 2/a\\ /a\
2*|1 - tan |-|| 4*tan|-|
\ \2// \2/
- --------------- + -----------
2/a\ 2/a\
1 + tan |-| 1 + tan |-|
\2/ \2/
$$- \frac{2 \cdot \left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} + \frac{4 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}$$
/a pi\ /a\
4*tan|- + --| 4*cot|-|
\2 4 / \2/
- ---------------- + -----------
2/a pi\ 2/a\
1 + tan |- + --| 1 + cot |-|
\2 4 / \2/
$$\frac{4 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} - \frac{4 \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1}$$
/a pi\ /a\
4*tan|- + --| 4*tan|-|
\2 4 / \2/
- ---------------- + -----------
2/a pi\ 2/a\
1 + tan |- + --| 1 + tan |-|
\2 4 / \2/
$$- \frac{4 \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} + \frac{4 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}$$
/ 1 \
2*|1 - -------|
| 2/a\|
| cot |-||
\ \2// 4
- --------------- + --------------------
1 / 1 \ /a\
1 + ------- |1 + -------|*cot|-|
2/a\ | 2/a\| \2/
cot |-| | cot |-||
\2/ \ \2//
$$- \frac{2 \cdot \left(1 - \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right)}{1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}} + \frac{4}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \cot{\left(\frac{a}{2} \right)}}$$
/ 2/a\\ / 2/a pi\\
2*|-1 + cot |-|| 2*|-1 + tan |- + --||
\ \2// \ \2 4 //
- ---------------- + ---------------------
2/a\ 2/a pi\
1 + cot |-| 1 + tan |- + --|
\2/ \2 4 /
$$\frac{2 \left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1\right)}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} - \frac{2 \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{a}{2} \right)} + 1}$$
// / pi\ \
|| 1 for |a + --| mod 2*pi = 0|
|| \ 4 / |
|| |
___ || 2/a pi\ |
-2*\/ 2 *|<-1 + cot |- + --| |
|| \2 8 / |
||----------------- otherwise |
|| 2/a pi\ |
|| 1 + cot |- + --| |
\\ \2 8 / /
$$- 2 \sqrt{2} \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
/ 2/a\\ / 2/a pi\\
2*|1 - tan |-|| 2*|1 - cot |- + --||
\ \2// \ \2 4 //
- --------------- + --------------------
2/a\ 2/a pi\
1 + tan |-| 1 + cot |- + --|
\2/ \2 4 /
$$- \frac{2 \cdot \left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} + \frac{2 \cdot \left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1}$$
// / pi\ \
|| 1 for |a + --| mod 2*pi = 0|
___ || \ 4 / |
-2*\/ 2 *|< |
|| 2/a pi\ / 2/a pi\\ |
||sin |- + --|*|-1 + cot |- + --|| otherwise |
\\ \2 8 / \ \2 8 // /
$$- 2 \sqrt{2} \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{\pi}{4}\right) \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} - 1\right) \sin^{2}{\left(\frac{a}{2} + \frac{\pi}{8} \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\ // 0 for a mod pi = 0\
- 2*|< | + 2*|< |
\\cos(a) otherwise / \\sin(a) otherwise /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) - \left(2 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for a mod 2*pi = 0\
|| | // 0 for a mod pi = 0\
- 2*|< / pi\ | + 2*|< |
||sin|a + --| otherwise | \\sin(a) otherwise /
\\ \ 2 / /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) - \left(2 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\sin{\left(a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for a mod pi = 0\
// 1 for a mod 2*pi = 0\ || |
- 2*|< | + 2*|< / pi\ |
\\cos(a) otherwise / ||cos|a - --| otherwise |
\\ \ 2 / /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) - \left(2 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right)\right)$$
2/a\
8*sin |-|*sin(a)
4*(-1 - cos(2*a) + 2*cos(a)) \2/
- ------------------------------ + -------------------
2 2 4/a\
1 - cos(2*a) + 2*(1 - cos(a)) sin (a) + 4*sin |-|
\2/
$$\frac{8 \sin^{2}{\left(\frac{a}{2} \right)} \sin{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)} + \sin^{2}{\left(a \right)}} - \frac{4 \cdot \left(2 \cos{\left(a \right)} - \cos{\left(2 a \right)} - 1\right)}{2 \left(- \cos{\left(a \right)} + 1\right)^{2} - \cos{\left(2 a \right)} + 1}$$
// / 3*pi\ \
// 1 for a mod 2*pi = 0\ || 1 for |a + ----| mod 2*pi = 0|
- 2*|< | + 2*|< \ 2 / |
\\cos(a) otherwise / || |
\\sin(a) otherwise /
$$\left(- 2 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for a mod pi = 0\
// 1 for a mod 2*pi = 0\ || |
|| | || 1 |
- 2*|< 1 | + 2*|<----------- otherwise |
||------ otherwise | || / pi\ |
\\sec(a) / ||sec|a - --| |
\\ \ 2 / /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(2 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(a \right)}} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for a mod 2*pi = 0\
|| | // 0 for a mod pi = 0\
|| 1 | || |
- 2*|<----------- otherwise | + 2*|< 1 |
|| /pi \ | ||------ otherwise |
||csc|-- - a| | \\csc(a) /
\\ \2 / /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\csc{\left(a \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(2 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for a mod pi = 0\
|| |
// 1 for a mod 2*pi = 0\ ||1 - cos(a) |
- 2*|< | + 2*|<---------- otherwise |
\\cos(a) otherwise / || /a\ |
|| tan|-| |
\\ \2/ /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{- \cos{\left(a \right)} + 1}{\tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(2 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right)\right)$$
/ 4/a\\
| 4*sin |-||
| \2/|
2*|1 - ---------| 2/a\
| 2 | 8*sin |-|
\ sin (a) / \2/
- ----------------- + ----------------------
4/a\ / 4/a\\
4*sin |-| | 4*sin |-||
\2/ | \2/|
1 + --------- |1 + ---------|*sin(a)
2 | 2 |
sin (a) \ sin (a) /
$$- \frac{2 \left(- \frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right)}{\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1} + \frac{8 \sin^{2}{\left(\frac{a}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right) \sin{\left(a \right)}}$$
// / pi\ \
|| 0 for |a + --| mod pi = 0|
|| \ 2 / | // 0 for a mod pi = 0\
- 2*|< | + 2*|< |
|| /a pi\ | \\sin(a) otherwise /
||(1 + sin(a))*cot|- + --| otherwise |
\\ \2 4 / /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) - \left(2 \left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(a \right)} + 1\right) \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for a mod 2*pi = 0\ // 0 for a mod pi = 0\
|| | || |
|| 2/a\ | || /a\ |
||-1 + cot |-| | || 2*cot|-| |
- 2*|< \2/ | + 2*|< \2/ |
||------------ otherwise | ||----------- otherwise |
|| 2/a\ | || 2/a\ |
||1 + cot |-| | ||1 + cot |-| |
\\ \2/ / \\ \2/ /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) - \left(2 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for a mod 2*pi = 0\ // 0 for a mod pi = 0\
|| | || |
|| 2/a\ | || /a\ |
||1 - tan |-| | || 2*tan|-| |
- 2*|< \2/ | + 2*|< \2/ |
||----------- otherwise | ||----------- otherwise |
|| 2/a\ | || 2/a\ |
||1 + tan |-| | ||1 + tan |-| |
\\ \2/ / \\ \2/ /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) - \left(2 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{a}{2} \right)} + 1}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for a mod 2*pi = 0\ // 0 for a mod pi = 0\
|| | || |
- 2*|
$$\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) - \left(2 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)$$
/ 2/a\ \
| sec |-| |
| \2/ |
2*|1 - ------------|
| 2/a pi\| /a\
| sec |- - --|| 4*sec|-|
\ \2 2 // \2/
- -------------------- + ------------------------------
2/a\ / 2/a\ \
sec |-| | sec |-| |
\2/ | \2/ | /a pi\
1 + ------------ |1 + ------------|*sec|- - --|
2/a pi\ | 2/a pi\| \2 2 /
sec |- - --| | sec |- - --||
\2 2 / \ \2 2 //
$$- \frac{2 \left(- \frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)}{\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1} + \frac{4 \sec{\left(\frac{a}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}$$
// 1 for a mod 2*pi = 0\
|| |
|| 1 | // 0 for a mod pi = 0\
||-1 + ------- | || |
|| 2/a\ | || 2 |
|| tan |-| | ||-------------------- otherwise |
- 2*|< \2/ | + 2*|
$$\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right) \tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(2 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right)$$
/ 2/a pi\\
| cos |- - --||
| \2 2 /|
2*|1 - ------------|
| 2/a\ | /a pi\
| cos |-| | 4*cos|- - --|
\ \2/ / \2 2 /
- -------------------- + -------------------------
2/a pi\ / 2/a pi\\
cos |- - --| | cos |- - --||
\2 2 / | \2 2 /| /a\
1 + ------------ |1 + ------------|*cos|-|
2/a\ | 2/a\ | \2/
cos |-| | cos |-| |
\2/ \ \2/ /
$$- \frac{2 \cdot \left(1 - \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right)}{1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}} + \frac{4 \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right) \cos{\left(\frac{a}{2} \right)}}$$
// / pi\ \
|| 0 for |a + --| mod pi = 0| // 0 for a mod pi = 0\
|| \ 2 / | || |
|| | || /a\ |
|| /a pi\ | || 2*cot|-| |
- 2*|< 2*cot|- + --| | + 2*|< \2/ |
|| \2 4 / | ||----------- otherwise |
||---------------- otherwise | || 2/a\ |
|| 2/a pi\ | ||1 + cot |-| |
||1 + cot |- + --| | \\ \2/ /
\\ \2 4 / /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) - \left(2 \left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
/ 2/pi a\\
| csc |-- - -||
| \2 2/|
2*|1 - ------------|
| 2/a\ | /pi a\
| csc |-| | 4*csc|-- - -|
\ \2/ / \2 2/
- -------------------- + -------------------------
2/pi a\ / 2/pi a\\
csc |-- - -| | csc |-- - -||
\2 2/ | \2 2/| /a\
1 + ------------ |1 + ------------|*csc|-|
2/a\ | 2/a\ | \2/
csc |-| | csc |-| |
\2/ \ \2/ /
$$- \frac{2 \cdot \left(1 - \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right)}{1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}} + \frac{4 \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right) \csc{\left(\frac{a}{2} \right)}}$$
// / 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*|< \2/ | + 2*|<-1 + tan |- + --| |
||------------ otherwise | || \2 4 / |
|| 2/a\ | ||----------------- otherwise |
||1 + cot |-| | || 2/a pi\ |
\\ \2/ / || 1 + tan |- + --| |
\\ \2 4 / /
$$\left(- 2 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for a mod pi = 0\
|| |
// 1 for a mod 2*pi = 0\ || 2*sin(a) |
|| | ||---------------------------- otherwise |
|| 2 | || / 2 \ |
- 2*|< -4 + 4*sin (a) + 4*cos(a) | + 2*|< | sin (a) | |
||--------------------------- otherwise | ||(1 - cos(a))*|1 + ---------| |
|| 2 2 | || | 4/a\| |
\\2*(1 - cos(a)) + 2*sin (a) / || | 4*sin |-|| |
|| \ \2// |
\\ /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \sin{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \left(- \cos{\left(a \right)} + 1\right)} & \text{otherwise} \end{cases}\right)\right) - \left(2 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{4 \sin^{2}{\left(a \right)} + 4 \cos{\left(a \right)} - 4}{2 \left(- \cos{\left(a \right)} + 1\right)^{2} + 2 \sin^{2}{\left(a \right)}} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for a mod 2*pi = 0\
|| |
|| 2 | // 0 for a mod pi = 0\
|| sin (a) | || |
||-1 + --------- | || sin(a) |
|| 4/a\ | ||----------------------- otherwise |
|| 4*sin |-| | ||/ 2 \ |
- 2*|< \2/ | + 2*|<| sin (a) | 2/a\ |
||-------------- otherwise | |||1 + ---------|*sin |-| |
|| 2 | ||| 4/a\| \2/ |
|| sin (a) | ||| 4*sin |-|| |
||1 + --------- | ||\ \2// |
|| 4/a\ | \\ /
|| 4*sin |-| |
\\ \2/ /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{\sin{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \sin^{2}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(2 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}}{1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for a mod 2*pi = 0\ // 0 for a mod pi = 0\
|| | || |
||/ 1 for a mod 2*pi = 0 | ||/ 0 for a mod pi = 0 |
||| | ||| |
||| 2/a\ | ||| /a\ |
- 2*|<|-1 + cot |-| | + 2*|<| 2*cot|-| |
||< \2/ otherwise | ||< \2/ otherwise |
|||------------ otherwise | |||----------- otherwise |
||| 2/a\ | ||| 2/a\ |
|||1 + cot |-| | |||1 + cot |-| |
\\\ \2/ / \\\ \2/ /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) - \left(2 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for a mod 2*pi = 0\
|| |
|| 2/a\ | // 0 for a mod pi = 0\
|| cos |-| | || |
|| \2/ | || /a\ |
||-1 + ------------ | || 2*cos|-| |
|| 2/a pi\ | || \2/ |
|| cos |- - --| | ||------------------------------ otherwise |
- 2*|< \2 2 / | + 2*|
$$\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{a}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(2 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for a mod 2*pi = 0\
|| |
|| 2/a pi\ | // 0 for a mod pi = 0\
|| sec |- - --| | || |
|| \2 2 / | || /a pi\ |
||-1 + ------------ | || 2*sec|- - --| |
|| 2/a\ | || \2 2 / |
|| sec |-| | ||------------------------- otherwise |
- 2*|< \2/ | + 2*|
$$\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right) \sec{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(2 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for a mod 2*pi = 0\
|| |
|| 2/a\ | // 0 for a mod pi = 0\
|| csc |-| | || |
|| \2/ | || /a\ |
||-1 + ------------ | || 2*csc|-| |
|| 2/pi a\ | || \2/ |
|| csc |-- - -| | ||------------------------------ otherwise |
- 2*|< \2 2/ | + 2*|
$$\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{a}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(2 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right)$$
-2*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)) + 2*Piecewise((0, Mod(a = pi, 0)), (2*csc(a/2)/((1 + csc(a/2)^2/csc(pi/2 - a/2)^2)*csc(pi/2 - a/2)), True))
Объединение рациональных выражений
[src]
2*(-cos(a) + sin(a))
$$2 \left(\sin{\left(a \right)} - \cos{\left(a \right)}\right)$$
2*(-cos(a) + sin(a))