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Упрощение выражений/ Общий знаменатель/ cos(pi/2+t)/(sin(pi-t)*tan(-t))

Общий знаменатель cos(pi/2+t)/(sin(pi-t)*tan(-t))

Выражение, которое надо упростить:

Решение

Вы ввели [src] 
       /pi    \    
    cos|-- + t|    
       \2     /    
-------------------
sin(pi - t)*tan(-t)
$$\frac{\cos{\left(t + \frac{\pi}{2} \right)}}{\sin{\left(- t + \pi \right)} \tan{\left(- t \right)}}$$ 
cos(pi/2 + t)/((sin(pi - t)*tan(-t)))
Разложение дроби [src] 
-1/tan(-t)
$$- \frac{1}{\tan{\left(- t \right)}}$$ 
  -1   
-------
tan(-t)
Общее упрощение [src] 
  1   
------
tan(t)
$$\frac{1}{\tan{\left(t \right)}}$$ 
1/tan(t)
Собрать выражение [src] 
cot(t)*csc(t)*sin(t)
$$\sin{\left(t \right)} \cot{\left(t \right)} \csc{\left(t \right)}$$ 
cot(t)*csc(t)*sin(t)
Раскрыть выражение [src] 
  1   
------
tan(t)
$$\frac{1}{\tan{\left(t \right)}}$$ 
1/tan(t)
Тригонометрическая часть [src] 
cot(t)
$$\cot{\left(t \right)}$$ 
  1   
------
tan(t)
$$\frac{1}{\tan{\left(t \right)}}$$ 
csc(t)
------
sec(t)
$$\frac{\csc{\left(t \right)}}{\sec{\left(t \right)}}$$ 
cos(t)
------
sin(t)
$$\frac{\cos{\left(t \right)}}{\sin{\left(t \right)}}$$ 
 sin(2*t)
---------
     2   
2*sin (t)
$$\frac{\sin{\left(2 t \right)}}{2 \sin^{2}{\left(t \right)}}$$ 
    2     
 csc (t)  
----------
2*csc(2*t)
$$\frac{\csc^{2}{\left(t \right)}}{2 \csc{\left(2 t \right)}}$$ 
   cos(t)  
-----------
   /    pi\
cos|t - --|
   \    2 /
$$\frac{\cos{\left(t \right)}}{\cos{\left(t - \frac{\pi}{2} \right)}}$$ 
   /    pi\
sec|t - --|
   \    2 /
-----------
   sec(t)
$$\frac{\sec{\left(t - \frac{\pi}{2} \right)}}{\sec{\left(t \right)}}$$ 
   /    pi\
sin|t + --|
   \    2 /
-----------
   sin(t)
$$\frac{\sin{\left(t + \frac{\pi}{2} \right)}}{\sin{\left(t \right)}}$$ 
   csc(t)  
-----------
   /pi    \
csc|-- - t|
   \2     /
$$\frac{\csc{\left(t \right)}}{\csc{\left(- t + \frac{\pi}{2} \right)}}$$ 
   /pi    \
sec|-- - t|
   \2     /
-----------
   sec(t)
$$\frac{\sec{\left(- t + \frac{\pi}{2} \right)}}{\sec{\left(t \right)}}$$ 
          2/t\
-1 + 2*cos |-|
           \2/
--------------
    sin(t)
$$\frac{2 \cos^{2}{\left(\frac{t}{2} \right)} - 1}{\sin{\left(t \right)}}$$ 
csc(pi - t)
-----------
   /pi    \
csc|-- - t|
   \2     /
$$\frac{\csc{\left(- t + \pi \right)}}{\csc{\left(- t + \frac{\pi}{2} \right)}}$$ 
       2/t\
1 - tan |-|
        \2/
-----------
       /t\ 
  2*tan|-| 
       \2/
$$\frac{- \tan^{2}{\left(\frac{t}{2} \right)} + 1}{2 \tan{\left(\frac{t}{2} \right)}}$$ 
              /t\
           tan|-|
   1          \2/
-------- - ------
     /t\     2   
2*tan|-|         
     \2/
$$- \frac{\tan{\left(\frac{t}{2} \right)}}{2} + \frac{1}{2 \tan{\left(\frac{t}{2} \right)}}$$ 
   /      pi\ 
cos|2*t - --| 
   \      2 / 
--------------
     2/    pi\
2*cos |t - --|
      \    2 /
$$\frac{\cos{\left(2 t - \frac{\pi}{2} \right)}}{2 \cos^{2}{\left(t - \frac{\pi}{2} \right)}}$$ 
     2/    pi\ 
  sec |t - --| 
      \    2 / 
---------------
     /      pi\
2*sec|2*t - --|
     \      2 /
$$\frac{\sec^{2}{\left(t - \frac{\pi}{2} \right)}}{2 \sec{\left(2 t - \frac{\pi}{2} \right)}}$$ 
              2        
 /       2/t\\         
 |1 + tan |-|| *tan(t) 
 \        \2//         
-----------------------
  /       2   \    2/t\
4*\1 + tan (t)/*tan |-|
                    \2/
$$\frac{\left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2} \tan{\left(t \right)}}{4 \left(\tan^{2}{\left(t \right)} + 1\right) \tan^{2}{\left(\frac{t}{2} \right)}}$$ 
/       2/t\\    /t   pi\
|1 + cot |-||*tan|- + --|
\        \2//    \2   4 /
-------------------------
/       2/t   pi\\    /t\
|1 + tan |- + --||*cot|-|
\        \2   4 //    \2/
$$\frac{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right) \tan{\left(\frac{t}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right) \cot{\left(\frac{t}{2} \right)}}$$ 
/       2/t   pi\\ /        2/t\\
|1 + tan |- + --||*|-1 + cot |-||
\        \2   4 // \         \2//
---------------------------------
/       2/t\\ /        2/t   pi\\
|1 + cot |-||*|-1 + tan |- + --||
\        \2// \         \2   4 //
$$\frac{\left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(\frac{t}{2} \right)} - 1\right)}{\left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} - 1\right) \left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)}$$ 
/       2/t   pi\\ /       2/t\\
|1 + cot |- + --||*|1 - tan |-||
\        \2   4 // \        \2//
--------------------------------
/       2/t\\ /       2/t   pi\\
|1 + tan |-||*|1 - cot |- + --||
\        \2// \        \2   4 //
$$\frac{\left(- \tan^{2}{\left(\frac{t}{2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)}{\left(- \cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)}$$ 
                                //  zoo    for t mod pi = 0\
                                ||                         |
//   0      for 2*t mod pi = 0\ ||   1                     |
|<                            |*|<-------     otherwise    |
\\sin(2*t)      otherwise     / ||   2                     |
                                ||sin (t)                  |
                                \\                         /
------------------------------------------------------------
                             2
$$\frac{\left(\begin{cases} 0 & \text{for}\: 2 t \bmod \pi = 0 \\\sin{\left(2 t \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\frac{1}{\sin^{2}{\left(t \right)}} & \text{otherwise} \end{cases}\right)}{2}$$ 
                              //                     /    3*pi\             \
                              ||       1         for |t + ----| mod 2*pi = 0|
                              ||                     \     2  /             |
                              ||                                            |
//  1     for t mod 2*pi = 0\ ||  1         /t\                             |
|<                          |*|<------ + tan|-|                             |
\\cos(t)      otherwise     / ||   /t\      \2/                             |
                              ||tan|-|                                      |
                              ||   \2/                                      |
                              ||---------------           otherwise         |
                              \\       2                                    /
$$\left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\cos{\left(t \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(t + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan{\left(\frac{t}{2} \right)} + \frac{1}{\tan{\left(\frac{t}{2} \right)}}}{2} & \text{otherwise} \end{cases}\right)$$ 
                                   //     zoo        for t mod pi = 0\
                                   ||                                |
//     0       for 2*t mod pi = 0\ ||             2                  |
||                               | ||/       2/t\\                   |
||  2*cot(t)                     | |||1 + cot |-||                   |
|<-----------      otherwise     |*|<\        \2//                   |
||       2                       | ||--------------     otherwise    |
||1 + cot (t)                    | ||       2/t\                     |
\\                               / ||  4*cot |-|                     |
                                   ||        \2/                     |
                                   \\                                /
----------------------------------------------------------------------
                                  2
$$\frac{\left(\begin{cases} 0 & \text{for}\: 2 t \bmod \pi = 0 \\\frac{2 \cot{\left(t \right)}}{\cot^{2}{\left(t \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)^{2}}{4 \cot^{2}{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}\right)}{2}$$ 
                                    //                       /    3*pi\             \
//     1        for t mod 2*pi = 0\ ||        1          for |t + ----| mod 2*pi = 0|
||                                | ||                       \     2  /             |
||        2/t\                    | ||                                              |
||-1 + cot |-|                    | ||        2/t   pi\                             |
|<         \2/                    |*|< 1 + tan |- + --|                             |
||------------      otherwise     | ||         \2   4 /                             |
||       2/t\                     | ||-----------------           otherwise         |
||1 + cot |-|                     | ||        2/t   pi\                             |
\\        \2/                     / ||-1 + tan |- + --|                             |
                                    \\         \2   4 /                             /
$$\left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{t}{2} \right)} - 1}{\cot^{2}{\left(\frac{t}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(t + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1}{\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} - 1} & \text{otherwise} \end{cases}\right)$$ 
Piecewise((1, Mod(t = 2*pi, 0)), ((-1 + cot(t/2)^2)/(1 + cot(t/2)^2), True))*Piecewise((1, Mod(t + 3*pi/2 = 2*pi, 0)), ((1 + tan(t/2 + pi/4)^2)/(-1 + tan(t/2 + pi/4)^2), True))
Объединение рациональных выражений [src] 
    /pi + 2*t\ 
-cos|--------| 
    \   2    / 
---------------
 sin(t)*tan(t)
$$- \frac{\cos{\left(\frac{2 t + \pi}{2} \right)}}{\sin{\left(t \right)} \tan{\left(t \right)}}$$ 
-cos((pi + 2*t)/2)/(sin(t)*tan(t))
Степени [src] 
  1   
------
tan(t)
$$\frac{1}{\tan{\left(t \right)}}$$ 
      -sin(t)      
-------------------
sin(pi - t)*tan(-t)
$$- \frac{\sin{\left(t \right)}}{\sin{\left(- t + \pi \right)} \tan{\left(- t \right)}}$$ 
  /   /    pi\      /     pi\\                
  | I*|t + --|    I*|-t - --||                
  |   \    2 /      \     2 /|                
  |e             e           | / I*t    -I*t\ 
2*|----------- + ------------|*\e    + e    / 
  \     2             2      /                
----------------------------------------------
/   I*(t - pi)    I*(pi - t)\ /   -I*t    I*t\
\- e           + e          /*\- e     + e   /
$$\frac{2 \left(e^{i t} + e^{- i t}\right) \left(\frac{e^{i \left(- t - \frac{\pi}{2}\right)}}{2} + \frac{e^{i \left(t + \frac{\pi}{2}\right)}}{2}\right)}{\left(e^{i t} - e^{- i t}\right) \left(e^{i \left(- t + \pi\right)} - e^{i \left(t - \pi\right)}\right)}$$ 
2*(exp(i*(t + pi/2))/2 + exp(i*(-t - pi/2))/2)*(exp(i*t) + exp(-i*t))/((-exp(i*(t - pi)) + exp(i*(pi - t)))*(-exp(-i*t) + exp(i*t)))
Численный ответ [src] 
cos(pi/2 + t)/(sin(pi - t)*tan(-t))
cos(pi/2 + t)/(sin(pi - t)*tan(-t))
Рациональный знаменатель [src] 
  1   
------
tan(t)
$$\frac{1}{\tan{\left(t \right)}}$$ 
1/tan(t)
Комбинаторика [src] 
  1   
------
tan(t)
$$\frac{1}{\tan{\left(t \right)}}$$ 
1/tan(t)
Общий знаменатель [src] 
  -1   
-------
tan(-t)
$$- \frac{1}{\tan{\left(- t \right)}}$$ 
-1/tan(-t)
    © Господин Экзамен – Калькулятор