Найти значение выражения 1-sin(a)^2+cot(a)^2*sin(a)^2еслиa=1 (1 минус синус от (a) в квадрате плюс котангенс от (a) в квадрате умножить на синус от (a) в квадрате еслиa равно 1) [Есть ответ!]
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1-sin(a)^2+cot(a)^2*sin(a)^2еслиa=1 (упростите выражение)

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Примеры

Решение

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