cot(a)^2*(cos(a)^2-1)+1еслиa=1/3 (упростите выражение)

Вы ввели

   2    /   2       \    
cot (a)*\cos (a) - 1/ + 1

$$\left(\cos^{2}{\left(a \right)} - 1\right) \cot^{2}{\left(a \right)} + 1$$

Подстановка условия

cot(a)^2*(cos(a)^2 - 1*1) + 1 при a = 1/3

подставляем

   2    /   2       \    
cot (a)*\cos (a) - 1/ + 1

$$\left(\cos^{2}{\left(a \right)} - 1\right) \cot^{2}{\left(a \right)} + 1$$

   2   
sin (a)

$$\sin^{2}{\left(a \right)}$$

переменные

a = 1/3

$$a = \frac{1}{3}$$

   2       
sin ((1/3))

$$\sin^{2}{\left((1/3) \right)}$$

   2     
sin (1/3)

$$\sin^{2}{\left(\frac{1}{3} \right)}$$

Степени

            /                   2\
            |     / I*a    -I*a\ |
       2    |     |e      e    | |
1 + cot (a)*|-1 + |---- + -----| |
            \     \ 2       2  / /

$$\left(\left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2}\right)^{2} - 1\right) \cot^{2}{\left(a \right)} + 1$$

Численный ответ

1.0 + cot(a)^2*(-1.0 + cos(a)^2)

Рациональный знаменатель

       2         2       2   
1 - cot (a) + cos (a)*cot (a)

$$\cos^{2}{\left(a \right)} \cot^{2}{\left(a \right)} - \cot^{2}{\left(a \right)} + 1$$

Общее упрощение

   2   
sin (a)

$$\sin^{2}{\left(a \right)}$$

Собрать выражение

       2    /  1   cos(2*a)\
1 + cot (a)*|- - + --------|
            \  2      2    /

$$\left(\frac{1}{2} \cos{\left (2 a \right )} - \frac{1}{2}\right) \cot^{2}{\left (a \right )} + 1$$

Комбинаторика

       2         2       2   
1 - cot (a) + cos (a)*cot (a)

$$\cos^{2}{\left(a \right)} \cot^{2}{\left(a \right)} - \cot^{2}{\left(a \right)} + 1$$

Общий знаменатель

       2         2       2   
1 - cot (a) + cos (a)*cot (a)

$$\cos^{2}{\left(a \right)} \cot^{2}{\left(a \right)} - \cot^{2}{\left(a \right)} + 1$$

Тригонометрическая часть

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

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

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

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

   2/    pi\
cos |a - --|
    \    2 /

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

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

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

/   0     for And(im(a) = 0, a mod pi = 0)
|                                         
<   2                                     
|sin (a)             otherwise            
\                                         

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

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

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

       2    /        2   \
    cos (a)*\-1 + cos (a)/
1 + ----------------------
            2/    pi\     
         cos |a - --|     
             \    2 /     

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

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

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

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

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

   2   
sin (a)

$$\sin^{2}{\left(a \right)}$$

                      2
         /       2/a\\ 
         |1 - tan |-|| 
         \        \2// 
    -1 + --------------
                      2
         /       2/a\\ 
         |1 + tan |-|| 
         \        \2// 
1 + -------------------
             2         
          tan (a)      

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

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

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

       2    /        2   \
    cos (a)*\-1 + cos (a)/
1 + ----------------------
              2           
           sin (a)        

$$\frac{\left(\cos^{2}{\left(a \right)} - 1\right) \cos^{2}{\left(a \right)}}{\sin^{2}{\left(a \right)}} + 1$$

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

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

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

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

       2      /        2   \
    sin (2*a)*\-1 + cos (a)/
1 + ------------------------
                4           
           4*sin (a)        

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

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

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

   1   
-------
   2   
csc (a)

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

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

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

1   cos(2*a)
- - --------
2      2    

$$\frac{1}{2} - \frac{\cos{\left(2 a \right)}}{2}$$

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

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

       2    /        1   \
    csc (a)*|-1 + -------|
            |        2   |
            \     sec (a)/
1 + ----------------------
              2           
           sec (a)        

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

     1      
------------
   2/    pi\
sec |a - --|
    \    2 /

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

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

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

       2/a\   
  4*tan |-|   
        \2/   
--------------
             2
/       2/a\\ 
|1 + tan |-|| 
\        \2// 

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

       2      /        2/    pi\\
    sin (2*a)*|-1 + sin |a + --||
              \         \    2 //
1 + -----------------------------
                   4             
              4*sin (a)          

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

Раскрыть выражение

       2         2       2   
1 - cot (a) + cos (a)*cot (a)

$$\cos^{2}{\left(a \right)} \cot^{2}{\left(a \right)} - \cot^{2}{\left(a \right)} + 1$$