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(1-cos(t))*(1+cos(t))еслиt=3 (упростите выражение)

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Примеры

Решение

Вы ввели [src]
(1 - cos(t))*(1 + cos(t))
$$\left(1 - \cos{\left(t \right)}\right) \left(\cos{\left(t \right)} + 1\right)$$
Подстановка условия [src]
(1 - cos(t))*(1 + cos(t)) при t = 3
подставляем
(1 - cos(t))*(1 + cos(t))
$$\left(1 - \cos{\left(t \right)}\right) \left(\cos{\left(t \right)} + 1\right)$$
   2   
sin (t)
$$\sin^{2}{\left(t \right)}$$
переменные
t = 3
$$t = 3$$
   2     
sin ((3))
$$\sin^{2}{\left((3) \right)}$$
   2   
sin (3)
$$\sin^{2}{\left(3 \right)}$$
Степени [src]
/     I*t    -I*t\ /     I*t    -I*t\
|    e      e    | |    e      e    |
|1 + ---- + -----|*|1 - ---- - -----|
\     2       2  / \     2       2  /
$$\left(- \frac{e^{i t}}{2} + 1 - \frac{e^{- i t}}{2}\right) \left(\frac{e^{i t}}{2} + 1 + \frac{e^{- i t}}{2}\right)$$
Численный ответ [src]
(1.0 - cos(t))*(1.0 + cos(t))
Рациональный знаменатель [src]
       2   
1 - cos (t)
$$1 - \cos^{2}{\left(t \right)}$$
Общее упрощение [src]
   2   
sin (t)
$$\sin^{2}{\left(t \right)}$$
Собрать выражение [src]
1   cos(2*t)
- - --------
2      2
$$- \frac{1}{2} \cos{\left (2 t \right )} + \frac{1}{2}$$
Общий знаменатель [src]
       2   
1 - cos (t)
$$1 - \cos^{2}{\left(t \right)}$$
Тригонометрическая часть [src]
    //                         1                           for And(im(t) = 0, t mod 2*pi = 0)\
    ||                                                                                       |
    ||    /     1        for And(im(t) = 0, t mod pi = 0)                                    |
    ||    |                                                                                  |
    ||    |        2                                                                         |
1 - |<    <-1 + cot (t)                                                                      |
    ||    |------------             otherwise                                                |
    ||    |       2                                                                          |
    ||1   \1 + cot (t)                                                                       |
    ||- + -----------------------------------------------              otherwise             |
    \\2                          2                                                           /
$$1 - \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge t \bmod 2 \pi = 0 \\\left(\frac{\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge t \bmod \pi = 0 \\\frac{\cot^{2}{\left(t \right)} - 1}{\cot^{2}{\left(t \right)} + 1} & \text{otherwise} \end{cases}}{2}\right) + \frac{1}{2} & \text{otherwise} \end{cases}\right)$$
1   cos(2*t)
- - --------
2      2
$$\frac{1}{2} - \frac{\cos{\left(2 t \right)}}{2}$$
    //                       1                         for And(im(t) = 0, t mod 2*pi = 0)\
    ||                                                                                   |
    ||    /   1      for And(im(t) = 0, t mod pi = 0)                                    |
1 - |<    <                                                                              |
    ||1   \cos(2*t)             otherwise                                                |
    ||- + -------------------------------------------              otherwise             |
    \\2                        2                                                         /
$$1 - \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge t \bmod 2 \pi = 0 \\\left(\frac{\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge t \bmod \pi = 0 \\\cos{\left(2 t \right)} & \text{otherwise} \end{cases}}{2}\right) + \frac{1}{2} & \text{otherwise} \end{cases}\right)$$
    //        1          for And(im(t) = 0, t mod 2*pi = 0)\
    ||                                                     |
    ||       /pi      \                                    |
1 - |<    sin|-- + 2*t|                                    |
    ||1      \2       /                                    |
    ||- + -------------              otherwise             |
    \\2         2                                          /
$$1 - \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge t \bmod 2 \pi = 0 \\\frac{\sin{\left(2 t + \frac{\pi}{2} \right)}}{2} + \frac{1}{2} & \text{otherwise} \end{cases}\right)$$
/           2/t\\ /           2/t\\
|    1 - tan |-|| |    1 - tan |-||
|            \2/| |            \2/|
|1 + -----------|*|1 - -----------|
|           2/t\| |           2/t\|
|    1 + tan |-|| |    1 + tan |-||
\            \2// \            \2//
$$\left(- \frac{1 - \tan^{2}{\left(\frac{t}{2} \right)}}{\tan^{2}{\left(\frac{t}{2} \right)} + 1} + 1\right) \left(\frac{1 - \tan^{2}{\left(\frac{t}{2} \right)}}{\tan^{2}{\left(\frac{t}{2} \right)} + 1} + 1\right)$$
     1      
------------
   2/    pi\
sec |t - --|
    \    2 /
$$\frac{1}{\sec^{2}{\left(t - \frac{\pi}{2} \right)}}$$
/   0     for And(im(t) = 0, t mod pi = 0)
|                                         
<   2                                     
|sin (t)             otherwise            
\
$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge t \bmod \pi = 0 \\\sin^{2}{\left(t \right)} & \text{otherwise} \end{cases}$$
   2/    pi\
cos |t - --|
    \    2 /
$$\cos^{2}{\left(t - \frac{\pi}{2} \right)}$$
    //   1     for And(im(t) = 0, t mod 2*pi = 0)\
    ||                                           |
1 - |<   2                                       |
    ||cos (t)              otherwise             |
    \\                                           /
$$1 - \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge t \bmod 2 \pi = 0 \\\cos^{2}{\left(t \right)} & \text{otherwise} \end{cases}\right)$$
   2   
sin (t)
$$\sin^{2}{\left(t \right)}$$
    //         1           for And(im(t) = 0, t mod 2*pi = 0)\
    ||                                                       |
    ||1          1                                           |
1 - |<- + ---------------              otherwise             |
    ||2        /pi      \                                    |
    ||    2*csc|-- - 2*t|                                    |
    \\         \2       /                                    /
$$1 - \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge t \bmod 2 \pi = 0 \\\frac{1}{2} + \frac{1}{2 \csc{\left(- 2 t + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
/      0         for And(im(t) = 0, t mod pi = 0)
|                                                
|       2/t\                                     
|  4*cot |-|                                     
|        \2/                                     
<--------------             otherwise            
|             2                                  
|/       2/t\\                                   
||1 + cot |-||                                   
|\        \2//                                   
\
$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge t \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{t}{2} \right)}}{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}$$
/    //     1        for And(im(t) = 0, t mod 2*pi = 0)\\ /    //     1        for And(im(t) = 0, t mod 2*pi = 0)\\
|    ||                                                || |    ||                                                ||
|    ||        2/t\                                    || |    ||        2/t\                                    ||
|    ||-1 + cot |-|                                    || |    ||-1 + cot |-|                                    ||
|1 - |<         \2/                                    ||*|1 + |<         \2/                                    ||
|    ||------------              otherwise             || |    ||------------              otherwise             ||
|    ||       2/t\                                     || |    ||       2/t\                                     ||
|    ||1 + cot |-|                                     || |    ||1 + cot |-|                                     ||
\    \\        \2/                                     // \    \\        \2/                                     //
$$\left(1 - \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge t \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{t}{2} \right)} - 1}{\cot^{2}{\left(\frac{t}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) \left(\left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge t \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{t}{2} \right)} - 1}{\cot^{2}{\left(\frac{t}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1\right)$$
    //         1           for And(im(t) = 0, t mod 2*pi = 0)\
    ||                                                       |
    ||             2                                         |
1 - |<1     1 - tan (t)                                      |
    ||- + ---------------              otherwise             |
    ||2     /       2   \                                    |
    \\    2*\1 + tan (t)/                                    /
$$1 - \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge t \bmod 2 \pi = 0 \\\frac{1 - \tan^{2}{\left(t \right)}}{2 \left(\tan^{2}{\left(t \right)} + 1\right)} + \frac{1}{2} & \text{otherwise} \end{cases}\right)$$
    //     1        for And(im(t) = 0, t mod 2*pi = 0)\
    ||                                                |
1 - |<1   cos(2*t)                                    |
    ||- + --------              otherwise             |
    \\2      2                                        /
$$1 - \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge t \bmod 2 \pi = 0 \\\frac{\cos{\left(2 t \right)}}{2} + \frac{1}{2} & \text{otherwise} \end{cases}\right)$$
/       /    pi\\ /       /    pi\\
|1 - sin|t + --||*|1 + sin|t + --||
\       \    2 // \       \    2 //
$$\left(1 - \sin{\left(t + \frac{\pi}{2} \right)}\right) \left(\sin{\left(t + \frac{\pi}{2} \right)} + 1\right)$$
/         1     \ /         1     \
|1 + -----------|*|1 - -----------|
|       /pi    \| |       /pi    \|
|    csc|-- - t|| |    csc|-- - t||
\       \2     // \       \2     //
$$\left(1 - \frac{1}{\csc{\left(- t + \frac{\pi}{2} \right)}}\right) \left(1 + \frac{1}{\csc{\left(- t + \frac{\pi}{2} \right)}}\right)$$
/      1   \ /      1   \
|1 + ------|*|1 - ------|
\    sec(t)/ \    sec(t)/
$$\left(1 - \frac{1}{\sec{\left(t \right)}}\right) \left(1 + \frac{1}{\sec{\left(t \right)}}\right)$$
   1   
-------
   2   
csc (t)
$$\frac{1}{\csc^{2}{\left(t \right)}}$$
       2/t\   
  4*tan |-|   
        \2/   
--------------
             2
/       2/t\\ 
|1 + tan |-|| 
\        \2//
$$\frac{4 \tan^{2}{\left(\frac{t}{2} \right)}}{\left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}}$$
    //      1         for And(im(t) = 0, t mod 2*pi = 0)\
    ||                                                  |
1 - |<1       1                                         |
    ||- + ----------              otherwise             |
    \\2   2*sec(2*t)                                    /
$$1 - \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(t\right)} = 0 \wedge t \bmod 2 \pi = 0 \\\frac{1}{2} + \frac{1}{2 \sec{\left(2 t \right)}} & \text{otherwise} \end{cases}\right)$$
Комбинаторика [src]
-(1 + cos(t))*(-1 + cos(t))
$$- \left(\cos{\left(t \right)} - 1\right) \left(\cos{\left(t \right)} + 1\right)$$
Раскрыть выражение [src]
       2   
1 - cos (t)
$$1 - \cos^{2}{\left(t \right)}$$