sin(a)^2+cos(a)^2+tan(b)^2еслиa=4 (упростите выражение)

Вы ввели

   2         2         2   
sin (a) + cos (a) + tan (b)

$$\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)} + \tan^{2}{\left(b \right)}$$

Подстановка условия

sin(a)^2 + cos(a)^2 + tan(b)^2 при a = 4

подставляем

   2         2         2   
sin (a) + cos (a) + tan (b)

$$\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)} + \tan^{2}{\left(b \right)}$$

   1   
-------
   2   
cos (b)

$$\frac{1}{\cos^{2}{\left(b \right)}}$$

переменные

a = 4

$$a = 4$$

   1   
-------
   2   
cos (b)

$$\frac{1}{\cos^{2}{\left(b \right)}}$$

Степени

              2                   2                   2
/ I*a    -I*a\    /   -I*a    I*a\    /   I*b    -I*b\ 
|e      e    |    \- e     + e   /    \- e    + e    / 
|---- + -----|  - ----------------- - -----------------
\ 2       2  /            4                          2 
                                       / I*b    -I*b\  
                                       \e    + e    /  

$$\left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2}\right)^{2} - \frac{\left(e^{i a} - e^{- i a}\right)^{2}}{4} - \frac{\left(- e^{i b} + e^{- i b}\right)^{2}}{\left(e^{i b} + e^{- i b}\right)^{2}}$$

Численный ответ

cos(a)^2 + sin(a)^2 + tan(b)^2

Общее упрощение

   1   
-------
   2   
cos (b)

$$\frac{1}{\cos^{2}{\left(b \right)}}$$

Собрать выражение

       2   
1 + tan (b)

$$\tan^{2}{\left (b \right )} + 1$$

Тригонометрическая часть

   2/pi    \
csc |-- - b|
    \2     /

$$\csc^{2}{\left(- b + \frac{\pi}{2} \right)}$$

          //      0         for And(im(a) = 0, a mod pi = 0)\   //      1         for And(im(a) = 0, a mod 2*pi = 0)\
          ||                                                |   ||                                                  |
          ||       2/a\                                     |   ||             2                                    |
          ||  4*tan |-|                                     |   ||/       2/a\\                                     |
   2      ||        \2/                                     |   |||1 - tan |-||                                     |
tan (b) + |<--------------             otherwise            | + |<\        \2//                                     |
          ||             2                                  |   ||--------------              otherwise             |
          ||/       2/a\\                                   |   ||             2                                    |
          |||1 + tan |-||                                   |   ||/       2/a\\                                     |
          ||\        \2//                                   |   |||1 + tan |-||                                     |
          \\                                                /   \\\        \2//                                     /

$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{4 \tan^{2}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\frac{\left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) + \tan^{2}{\left(b \right)}$$

   2/pi    \   //   0     for And(im(a) = 0, a mod pi = 0)\   //     1        for And(im(a) = 0, a mod 2*pi = 0)\
csc |-- - b|   ||                                         |   ||                                                |
    \2     /   ||   1                                     |   ||     1                                          |
------------ + |<-------             otherwise            | + |<------------              otherwise             |
     2         ||   2                                     |   ||   2/pi    \                                    |
  csc (b)      ||csc (a)                                  |   ||csc |-- - a|                                    |
               \\                                         /   \\    \2     /                                    /

$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{1}{\csc^{2}{\left(a \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\frac{1}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \frac{\csc^{2}{\left(- b + \frac{\pi}{2} \right)}}{\csc^{2}{\left(b \right)}}$$

          //   0     for And(im(a) = 0, a mod pi = 0)\   //   1     for And(im(a) = 0, a mod 2*pi = 0)\
   2      ||                                         |   ||                                           |
tan (b) + |<   2                                     | + |<   2                                       |
          ||sin (a)             otherwise            |   ||cos (a)              otherwise             |
          \\                                         /   \\                                           /

$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\sin^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\cos^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right) + \tan^{2}{\left(b \right)}$$

                              2      
   1           1           sec (b)   
------- + ------------ + ------------
   2         2/    pi\      2/    pi\
sec (a)   sec |a - --|   sec |b - --|
              \    2 /       \    2 /

$$\frac{\sec^{2}{\left(b \right)}}{\sec^{2}{\left(b - \frac{\pi}{2} \right)}} + \frac{1}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} + \frac{1}{\sec^{2}{\left(a \right)}}$$

     1      
------------
   2/    pi\
sin |b + --|
    \    2 /

$$\frac{1}{\sin^{2}{\left(b + \frac{\pi}{2} \right)}}$$

               //     0        for And(im(a) = 0, a mod pi = 0)\   //   1     for And(im(a) = 0, a mod 2*pi = 0)\
     2         ||                                              |   ||                                           |
  sec (b)      ||     1                                        |   ||   1                                       |
------------ + |<------------             otherwise            | + |<-------              otherwise             |
   2/    pi\   ||   2/    pi\                                  |   ||   2                                       |
sec |b - --|   ||sec |a - --|                                  |   ||sec (a)                                    |
    \    2 /   \\    \    2 /                                  /   \\                                           /

$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{1}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\frac{1}{\sec^{2}{\left(a \right)}} & \text{otherwise} \end{cases}\right) + \frac{\sec^{2}{\left(b \right)}}{\sec^{2}{\left(b - \frac{\pi}{2} \right)}}$$

/     1        for And(im(b) = 0, b mod 2*pi = 0)
|                                                
<     2                                          
|------------              otherwise             
\1 + cos(2*b)                                    

$$\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(b\right)} = 0 \wedge b \bmod 2 \pi = 0 \\\frac{2}{\cos{\left(2 b \right)} + 1} & \text{otherwise} \end{cases}$$

   1   
-------
   2   
cos (b)

$$\frac{1}{\cos^{2}{\left(b \right)}}$$

          //      0         for And(im(a) = 0, a mod pi = 0)\   //       1         for And(im(a) = 0, a mod 2*pi = 0)\
          ||                                                |   ||                                                   |
          ||       2/a\                                     |   ||              2                                    |
          ||  4*cot |-|                                     |   ||/        2/a\\                                     |
   1      ||        \2/                                     |   |||-1 + cot |-||                                     |
------- + |<--------------             otherwise            | + |<\         \2//                                     |
   2      ||             2                                  |   ||---------------              otherwise             |
cot (b)   ||/       2/a\\                                   |   ||              2                                    |
          |||1 + cot |-||                                   |   || /       2/a\\                                     |
          ||\        \2//                                   |   || |1 + cot |-||                                     |
          \\                                                /   \\ \        \2//                                     /

$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) + \frac{1}{\cot^{2}{\left(b \right)}}$$

                         4   
   2         2      4*sin (b)
cos (a) + sin (a) + ---------
                       2     
                    sin (2*b)

$$\sin^{2}{\left(a \right)} + \frac{4 \sin^{4}{\left(b \right)}}{\sin^{2}{\left(2 b \right)}} + \cos^{2}{\left(a \right)}$$

     4      //   0     for And(im(a) = 0, a mod pi = 0)\   //     1        for And(im(a) = 0, a mod 2*pi = 0)\
4*sin (b)   ||                                         |   ||                                                |
--------- + |<   2                                     | + |<   2/    pi\                                    |
   2        ||sin (a)             otherwise            |   ||sin |a + --|              otherwise             |
sin (2*b)   \\                                         /   \\    \    2 /                                    /

$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\sin^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\sin^{2}{\left(a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + \frac{4 \sin^{4}{\left(b \right)}}{\sin^{2}{\left(2 b \right)}}$$

     2      
------------
1 + cos(2*b)

$$\frac{2}{\cos{\left(2 b \right)} + 1}$$

/       1         for And(im(b) = 0, b mod 2*pi = 0)
|                                                   
|              2                                    
| /       2/b\\                                     
| |1 + cot |-||                                     
< \        \2//                                     
|---------------              otherwise             
|              2                                    
|/        2/b\\                                     
||-1 + cot |-||                                     
\\         \2//                                     

$$\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(b\right)} = 0 \wedge b \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{b}{2} \right)} + 1\right)^{2}}{\left(\cot^{2}{\left(\frac{b}{2} \right)} - 1\right)^{2}} & \text{otherwise} \end{cases}$$

                            2/pi    \
                         csc |-- - b|
   1           1             \2     /
------- + ------------ + ------------
   2         2/pi    \        2      
csc (a)   csc |-- - a|     csc (b)   
              \2     /               

$$\frac{1}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} + \frac{\csc^{2}{\left(- b + \frac{\pi}{2} \right)}}{\csc^{2}{\left(b \right)}} + \frac{1}{\csc^{2}{\left(a \right)}}$$

          //                        0                          for And(im(a) = 0, a mod pi = 0)\   //                         1                            for And(im(a) = 0, a mod 2*pi = 0)\
          ||                                                                                   |   ||                                                                                        |
          ||/      0         for And(im(a) = 0, a mod pi = 0)                                  |   ||/       1         for And(im(a) = 0, a mod 2*pi = 0)                                    |
          |||                                                                                  |   |||                                                                                       |
          |||       2/a\                                                                       |   |||              2                                                                        |
   1      |||  4*cot |-|                                                                       |   |||/        2/a\\                                                                         |
------- + |<|        \2/                                                                       | + |<||-1 + cot |-||                                                                         |
   2      ||<--------------             otherwise                         otherwise            |   ||<\         \2//                                                   otherwise             |
cot (b)   |||             2                                                                    |   |||---------------              otherwise                                                 |
          |||/       2/a\\                                                                     |   |||              2                                                                        |
          ||||1 + cot |-||                                                                     |   ||| /       2/a\\                                                                         |
          |||\        \2//                                                                     |   ||| |1 + cot |-||                                                                         |
          \\\                                                                                  /   \\\ \        \2//                                                                         /

$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \frac{1}{\cot^{2}{\left(b \right)}}$$

          //                    0                       for And(im(a) = 0, a mod pi = 0)\   //                     1                        for And(im(a) = 0, a mod 2*pi = 0)\
          ||                                                                            |   ||                                                                                |
   2      ||/   0     for And(im(a) = 0, a mod pi = 0)                                  |   ||/   1     for And(im(a) = 0, a mod 2*pi = 0)                                    |
tan (b) + |<|                                                                           | + |<|                                                                               |
          ||<   2                                                  otherwise            |   ||<   2                                                     otherwise             |
          |||sin (a)             otherwise                                              |   |||cos (a)              otherwise                                                 |
          \\\                                                                           /   \\\                                                                               /

$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\sin^{2}{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\cos^{2}{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \tan^{2}{\left(b \right)}$$

                              4   
   2         2/    pi\   4*sin (b)
sin (a) + sin |a + --| + ---------
              \    2 /      2     
                         sin (2*b)

$$\sin^{2}{\left(a \right)} + \frac{4 \sin^{4}{\left(b \right)}}{\sin^{2}{\left(2 b \right)}} + \sin^{2}{\left(a + \frac{\pi}{2} \right)}$$

                            2/    pi\
                         cos |b - --|
   2         2/    pi\       \    2 /
cos (a) + cos |a - --| + ------------
              \    2 /        2      
                           cos (b)   

$$\cos^{2}{\left(a \right)} + \cos^{2}{\left(a - \frac{\pi}{2} \right)} + \frac{\cos^{2}{\left(b - \frac{\pi}{2} \right)}}{\cos^{2}{\left(b \right)}}$$

   2/    pi\                                                                                                     
cos |b - --|   //     0        for And(im(a) = 0, a mod pi = 0)\   //   1     for And(im(a) = 0, a mod 2*pi = 0)\
    \    2 /   ||                                              |   ||                                           |
------------ + |<   2/    pi\                                  | + |<   2                                       |
     2         ||cos |a - --|             otherwise            |   ||cos (a)              otherwise             |
  cos (b)      \\    \    2 /                                  /   \\                                           /

$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\cos^{2}{\left(a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\cos^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right) + \frac{\cos^{2}{\left(b - \frac{\pi}{2} \right)}}{\cos^{2}{\left(b \right)}}$$

                       2                 
          /       2/a\\           2/a\   
          |1 - tan |-||      4*tan |-|   
   2      \        \2//            \2/   
tan (b) + -------------- + --------------
                       2                2
          /       2/a\\    /       2/a\\ 
          |1 + tan |-||    |1 + tan |-|| 
          \        \2//    \        \2// 

$$\frac{\left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} + \tan^{2}{\left(b \right)} + \frac{4 \tan^{2}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}$$

                       2   
   2         2      sin (b)
cos (a) + sin (a) + -------
                       2   
                    cos (b)

$$\sin^{2}{\left(a \right)} + \frac{\sin^{2}{\left(b \right)}}{\cos^{2}{\left(b \right)}} + \cos^{2}{\left(a \right)}$$

                       2   
   1         1      sec (b)
------- + ------- + -------
   2         2         2   
csc (a)   sec (a)   csc (b)

$$\frac{1}{\sec^{2}{\left(a \right)}} + \frac{\sec^{2}{\left(b \right)}}{\csc^{2}{\left(b \right)}} + \frac{1}{\csc^{2}{\left(a \right)}}$$

             2
/       2/b\\ 
|1 + tan |-|| 
\        \2// 
--------------
             2
/       2/b\\ 
|1 - tan |-|| 
\        \2// 

$$\frac{\left(\tan^{2}{\left(\frac{b}{2} \right)} + 1\right)^{2}}{\left(1 - \tan^{2}{\left(\frac{b}{2} \right)}\right)^{2}}$$

   2   
sec (b)

$$\sec^{2}{\left(b \right)}$$