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