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