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