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sin(7*y)^2-1 если y=1/4

Выражение, которое надо упростить:

Решение

Вы ввели [src]
   2         
sin (7*y) - 1
$$\sin^{2}{\left(7 y \right)} - 1$$
sin(7*y)^2 - 1*1
Общее упрощение [src]
    2     
-cos (7*y)
$$- \cos^{2}{\left(7 y \right)}$$
-cos(7*y)^2
Подстановка условия [src]
sin(7*y)^2 - 1*1 при y = 1/4
подставляем
   2         
sin (7*y) - 1
$$\sin^{2}{\left(7 y \right)} - 1$$
    2     
-cos (7*y)
$$- \cos^{2}{\left(7 y \right)}$$
переменные
y = 1/4
$$y = \frac{1}{4}$$
    2         
-cos (7*(1/4))
$$- \cos^{2}{\left(7 (1/4) \right)}$$
    2     
-cos (7/4)
$$- \cos^{2}{\left(\frac{7}{4} \right)}$$
-cos(7/4)^2
Численный ответ [src]
-1.0 + sin(7*y)^2
-1.0 + sin(7*y)^2
Степени [src]
                         2
     /   -7*I*y    7*I*y\ 
     \- e       + e     / 
-1 - ---------------------
               4          
$$- \frac{\left(e^{7 i y} - e^{- 7 i y}\right)^{2}}{4} - 1$$
-1 - (-exp(-7*i*y) + exp(7*i*y))^2/4
Комбинаторика [src]
(1 + sin(7*y))*(-1 + sin(7*y))
$$\left(\sin{\left(7 y \right)} - 1\right) \left(\sin{\left(7 y \right)} + 1\right)$$
(1 + sin(7*y))*(-1 + sin(7*y))
Раскрыть выражение [src]
                                                                             2
     /     7            4       3           6                   2       5   \ 
-1 + \- sin (y) - 35*cos (y)*sin (y) + 7*cos (y)*sin(y) + 21*cos (y)*sin (y)/ 
$$\left(- \sin^{7}{\left(y \right)} + 21 \sin^{5}{\left(y \right)} \cos^{2}{\left(y \right)} - 35 \sin^{3}{\left(y \right)} \cos^{4}{\left(y \right)} + 7 \sin{\left(y \right)} \cos^{6}{\left(y \right)}\right)^{2} - 1$$
              12               8             4            2              14              6               10   
-1 - 14336*sin  (y) - 13440*sin (y) - 784*sin (y) + 49*sin (y) + 4096*sin  (y) + 4704*sin (y) + 19712*sin  (y)
$$4096 \sin^{14}{\left(y \right)} - 14336 \sin^{12}{\left(y \right)} + 19712 \sin^{10}{\left(y \right)} - 13440 \sin^{8}{\left(y \right)} + 4704 \sin^{6}{\left(y \right)} - 784 \sin^{4}{\left(y \right)} + 49 \sin^{2}{\left(y \right)} - 1$$
-1 - 14336*sin(y)^12 - 13440*sin(y)^8 - 784*sin(y)^4 + 49*sin(y)^2 + 4096*sin(y)^14 + 4704*sin(y)^6 + 19712*sin(y)^10
Собрать выражение [src]
  1   cos(14*y)
- - - ---------
  2       2    
$$- \frac{\cos{\left(14 y \right)}}{2} - \frac{1}{2}$$
-1/2 - cos(14*y)/2
Тригонометрическая часть [src]
    2     
-cos (7*y)
$$- \cos^{2}{\left(7 y \right)}$$
   -1    
---------
   2     
sec (7*y)
$$- \frac{1}{\sec^{2}{\left(7 y \right)}}$$
         1    
-1 + ---------
        2     
     csc (7*y)
$$-1 + \frac{1}{\csc^{2}{\left(7 y \right)}}$$
    2/pi      \
-sin |-- + 7*y|
     \2       /
$$- \sin^{2}{\left(7 y + \frac{\pi}{2} \right)}$$
        2/      pi\
-1 + cos |7*y - --|
         \      2 /
$$\cos^{2}{\left(7 y - \frac{\pi}{2} \right)} - 1$$
  1   cos(14*y)
- - - ---------
  2       2    
$$- \frac{\cos{\left(14 y \right)}}{2} - \frac{1}{2}$$
           1       
-1 + --------------
        2          
     csc (pi - 7*y)
$$-1 + \frac{1}{\csc^{2}{\left(- 7 y + \pi \right)}}$$
     -1       
--------------
   2/pi      \
csc |-- - 7*y|
    \2       /
$$- \frac{1}{\csc^{2}{\left(- 7 y + \frac{\pi}{2} \right)}}$$
           1       
-1 + --------------
        2/      pi\
     sec |7*y - --|
         \      2 /
$$-1 + \frac{1}{\sec^{2}{\left(7 y - \frac{\pi}{2} \right)}}$$
           1       
-1 + --------------
        2/pi      \
     sec |-- - 7*y|
         \2       /
$$-1 + \frac{1}{\sec^{2}{\left(- 7 y + \frac{\pi}{2} \right)}}$$
         2           2     
  1   sin (7*y)   cos (7*y)
- - + --------- - ---------
  2       2           2    
$$\frac{\sin^{2}{\left(7 y \right)}}{2} - \frac{\cos^{2}{\left(7 y \right)}}{2} - \frac{1}{2}$$
            2/7*y\   
       4*cot |---|   
             \ 2 /   
-1 + ----------------
                    2
     /       2/7*y\\ 
     |1 + cot |---|| 
     \        \ 2 // 
$$-1 + \frac{4 \cot^{2}{\left(\frac{7 y}{2} \right)}}{\left(\cot^{2}{\left(\frac{7 y}{2} \right)} + 1\right)^{2}}$$
            2/7*y\   
       4*tan |---|   
             \ 2 /   
-1 + ----------------
                    2
     /       2/7*y\\ 
     |1 + tan |---|| 
     \        \ 2 // 
$$-1 + \frac{4 \tan^{2}{\left(\frac{7 y}{2} \right)}}{\left(\tan^{2}{\left(\frac{7 y}{2} \right)} + 1\right)^{2}}$$
 //    1      for 7*y mod 2*pi = 0\
 ||                               |
-|<   2                           |
 ||cos (7*y)       otherwise      |
 \\                               /
$$- \begin{cases} 1 & \text{for}\: 7 y \bmod 2 \pi = 0 \\\cos^{2}{\left(7 y \right)} & \text{otherwise} \end{cases}$$
                 4             
-1 + --------------------------
                    2          
     /        1    \     2/7*y\
     |1 + ---------| *cot |---|
     |       2/7*y\|      \ 2 /
     |    cot |---||           
     \        \ 2 //           
$$-1 + \frac{4}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{7 y}{2} \right)}}\right)^{2} \cot^{2}{\left(\frac{7 y}{2} \right)}}$$
     //    0      for 7*y mod pi = 0\
     ||                             |
-1 + |<   2                         |
     ||sin (7*y)      otherwise     |
     \\                             /
$$\left(\begin{cases} 0 & \text{for}\: 7 y \bmod \pi = 0 \\\sin^{2}{\left(7 y \right)} & \text{otherwise} \end{cases}\right) - 1$$
                2 
 /       2/7*y\\  
-|1 - tan |---||  
 \        \ 2 //  
------------------
                2 
 /       2/7*y\\  
 |1 + tan |---||  
 \        \ 2 //  
$$- \frac{\left(- \tan^{2}{\left(\frac{7 y}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{7 y}{2} \right)} + 1\right)^{2}}$$
     //    0      for 7*y mod pi = 0\
     ||                             |
     ||    1                        |
-1 + |<---------      otherwise     |
     ||   2                         |
     ||csc (7*y)                    |
     \\                             /
$$\left(\begin{cases} 0 & \text{for}\: 7 y \bmod \pi = 0 \\\frac{1}{\csc^{2}{\left(7 y \right)}} & \text{otherwise} \end{cases}\right) - 1$$
     //      0         for 7*y mod pi = 0\
     ||                                  |
-1 + |<   2/      pi\                    |
     ||cos |7*y - --|      otherwise     |
     \\    \      2 /                    /
$$\left(\begin{cases} 0 & \text{for}\: 7 y \bmod \pi = 0 \\\cos^{2}{\left(7 y - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) - 1$$
     //      0         for 7*y mod pi = 0\
     ||                                  |
     ||      1                           |
-1 + |<--------------      otherwise     |
     ||   2/      pi\                    |
     ||sec |7*y - --|                    |
     \\    \      2 /                    /
$$\left(\begin{cases} 0 & \text{for}\: 7 y \bmod \pi = 0 \\\frac{1}{\sec^{2}{\left(7 y - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) - 1$$
                          2
     /        2/pi   7*y\\ 
     |-1 + tan |-- + ---|| 
     \         \4     2 // 
-1 + ----------------------
                         2 
     /       2/pi   7*y\\  
     |1 + tan |-- + ---||  
     \        \4     2 //  
$$\frac{\left(\tan^{2}{\left(\frac{7 y}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{7 y}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} - 1$$
                         2
     /       2/pi   7*y\\ 
     |1 - cot |-- + ---|| 
     \        \4     2 // 
-1 + ---------------------
                         2
     /       2/pi   7*y\\ 
     |1 + cot |-- + ---|| 
     \        \4     2 // 
$$\frac{\left(- \cot^{2}{\left(\frac{7 y}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}{\left(\cot^{2}{\left(\frac{7 y}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} - 1$$
             2         4/7*y\  
       16*sin (7*y)*sin |---|  
                        \ 2 /  
-1 + --------------------------
                              2
     /   2             4/7*y\\ 
     |sin (7*y) + 4*sin |---|| 
     \                  \ 2 // 
$$\frac{16 \sin^{4}{\left(\frac{7 y}{2} \right)} \sin^{2}{\left(7 y \right)}}{\left(4 \sin^{4}{\left(\frac{7 y}{2} \right)} + \sin^{2}{\left(7 y \right)}\right)^{2}} - 1$$
                   4/7*y\        
             16*sin |---|        
                    \ 2 /        
-1 + ----------------------------
                      2          
     /         4/7*y\\           
     |    4*sin |---||           
     |          \ 2 /|     2     
     |1 + -----------| *sin (7*y)
     |        2      |           
     \     sin (7*y) /           
$$-1 + \frac{16 \sin^{4}{\left(\frac{7 y}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{7 y}{2} \right)}}{\sin^{2}{\left(7 y \right)}} + 1\right)^{2} \sin^{2}{\left(7 y \right)}}$$
     //       0          for 7*y mod pi = 0\
     ||                                    |
     ||       2/7*y\                       |
     ||  4*tan |---|                       |
     ||        \ 2 /                       |
-1 + |<----------------      otherwise     |
     ||               2                    |
     ||/       2/7*y\\                     |
     |||1 + tan |---||                     |
     ||\        \ 2 //                     |
     \\                                    /
$$\left(\begin{cases} 0 & \text{for}\: 7 y \bmod \pi = 0 \\\frac{4 \tan^{2}{\left(\frac{7 y}{2} \right)}}{\left(\tan^{2}{\left(\frac{7 y}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) - 1$$
     //       0          for 7*y mod pi = 0\
     ||                                    |
     ||       2/7*y\                       |
     ||  4*cot |---|                       |
     ||        \ 2 /                       |
-1 + |<----------------      otherwise     |
     ||               2                    |
     ||/       2/7*y\\                     |
     |||1 + cot |---||                     |
     ||\        \ 2 //                     |
     \\                                    /
$$\left(\begin{cases} 0 & \text{for}\: 7 y \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{7 y}{2} \right)}}{\left(\cot^{2}{\left(\frac{7 y}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) - 1$$
     //              0                 for 7*y mod pi = 0\
     ||                                                  |
     ||/    0      for 7*y mod pi = 0                    |
-1 + |<|                                                 |
     ||<   2                               otherwise     |
     |||sin (7*y)      otherwise                         |
     \\\                                                 /
$$\left(\begin{cases} 0 & \text{for}\: 7 y \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 7 y \bmod \pi = 0 \\\sin^{2}{\left(7 y \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) - 1$$
 //        1          for 7*y mod 2*pi = 0\
 ||                                       |
 ||                2                      |
 ||/        2/7*y\\                       |
 |||-1 + cot |---||                       |
-|<\         \ 2 //                       |
 ||-----------------       otherwise      |
 ||                2                      |
 || /       2/7*y\\                       |
 || |1 + cot |---||                       |
 \\ \        \ 2 //                       /
$$- \begin{cases} 1 & \text{for}\: 7 y \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{7 y}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{7 y}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}$$
     //                                 /      3*pi\             \
     ||             1               for |7*y + ----| mod 2*pi = 0|
     ||                                 \       2  /             |
-1 + |<                                                          |
     ||       4/7*y\        2/7*y\                               |
     ||- 4*cos |---| + 4*cos |---|            otherwise          |
     \\        \ 2 /         \ 2 /                               /
$$\left(\begin{cases} 1 & \text{for}\: \left(7 y + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\- 4 \cos^{4}{\left(\frac{7 y}{2} \right)} + 4 \cos^{2}{\left(\frac{7 y}{2} \right)} & \text{otherwise} \end{cases}\right) - 1$$
     //            0               for 7*y mod pi = 0\
     ||                                              |
     ||            4                                 |
     ||--------------------------      otherwise     |
     ||               2                              |
-1 + |
            
$$\left(\begin{cases} 0 & \text{for}\: 7 y \bmod \pi = 0 \\\frac{4}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{7 y}{2} \right)}}\right)^{2} \tan^{2}{\left(\frac{7 y}{2} \right)}} & \text{otherwise} \end{cases}\right) - 1$$
                        2/7*y\               
                   4*sec |---|               
                         \ 2 /               
-1 + ----------------------------------------
                           2                 
     /          2/7*y\    \                  
     |       sec |---|    |                  
     |           \ 2 /    |     2/  pi   7*y\
     |1 + ----------------| *sec |- -- + ---|
     |       2/  pi   7*y\|      \  2     2 /
     |    sec |- -- + ---||                  
     \        \  2     2 //                  
$$-1 + \frac{4 \sec^{2}{\left(\frac{7 y}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{7 y}{2} \right)}}{\sec^{2}{\left(\frac{7 y}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec^{2}{\left(\frac{7 y}{2} - \frac{\pi}{2} \right)}}$$
                  2/  pi   7*y\       
             4*cos |- -- + ---|       
                   \  2     2 /       
-1 + ---------------------------------
                           2          
     /       2/  pi   7*y\\           
     |    cos |- -- + ---||           
     |        \  2     2 /|     2/7*y\
     |1 + ----------------| *cos |---|
     |          2/7*y\    |      \ 2 /
     |       cos |---|    |           
     \           \ 2 /    /           
$$-1 + \frac{4 \cos^{2}{\left(\frac{7 y}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{7 y}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{7 y}{2} \right)}}\right)^{2} \cos^{2}{\left(\frac{7 y}{2} \right)}}$$
                  2/pi   7*y\       
             4*csc |-- - ---|       
                   \2     2 /       
-1 + -------------------------------
                         2          
     /       2/pi   7*y\\           
     |    csc |-- - ---||           
     |        \2     2 /|     2/7*y\
     |1 + --------------| *csc |---|
     |         2/7*y\   |      \ 2 /
     |      csc |---|   |           
     \          \ 2 /   /           
$$-1 + \frac{4 \csc^{2}{\left(- \frac{7 y}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{7 y}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{7 y}{2} \right)}}\right)^{2} \csc^{2}{\left(\frac{7 y}{2} \right)}}$$
     //             0                for 7*y mod pi = 0\
     ||                                                |
     ||            2                                   |
     ||         sin (7*y)                              |
     ||----------------------------      otherwise     |
     ||                 2                              |
-1 + |
            
$$\left(\begin{cases} 0 & \text{for}\: 7 y \bmod \pi = 0 \\\frac{\sin^{2}{\left(7 y \right)}}{\left(1 + \frac{\sin^{2}{\left(7 y \right)}}{4 \sin^{4}{\left(\frac{7 y}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{7 y}{2} \right)}} & \text{otherwise} \end{cases}\right) - 1$$
     //                  0                    for 7*y mod pi = 0\
     ||                                                         |
     ||/       0          for 7*y mod pi = 0                    |
     |||                                                        |
     |||       2/7*y\                                           |
     |||  4*cot |---|                                           |
-1 + |<|        \ 2 /                                           |
     ||<----------------      otherwise           otherwise     |
     |||               2                                        |
     |||/       2/7*y\\                                         |
     ||||1 + cot |---||                                         |
     |||\        \ 2 //                                         |
     \\\                                                        /
$$\left(\begin{cases} 0 & \text{for}\: 7 y \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 7 y \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{7 y}{2} \right)}}{\left(\cot^{2}{\left(\frac{7 y}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) - 1$$
     //                            /      3*pi\             \
     ||          1             for |7*y + ----| mod 2*pi = 0|
     ||                            \       2  /             |
     ||                                                     |
     ||                     2                               |
     ||/        2/pi   7*y\\                                |
-1 + |<|-1 + tan |-- + ---||                                |
     ||\         \4     2 //                                |
     ||----------------------            otherwise          |
     ||                    2                                |
     ||/       2/pi   7*y\\                                 |
     |||1 + tan |-- + ---||                                 |
     \\\        \4     2 //                                 /
$$\left(\begin{cases} 1 & \text{for}\: \left(7 y + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\left(\tan^{2}{\left(\frac{7 y}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{7 y}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) - 1$$
     //                0                  for 7*y mod pi = 0\
     ||                                                     |
     ||             2/  pi   7*y\                           |
     ||        4*sec |- -- + ---|                           |
     ||              \  2     2 /                           |
     ||---------------------------------      otherwise     |
     ||                      2                              |
-1 + |
            
$$\left(\begin{cases} 0 & \text{for}\: 7 y \bmod \pi = 0 \\\frac{4 \sec^{2}{\left(\frac{7 y}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{7 y}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{7 y}{2} \right)}}\right)^{2} \sec^{2}{\left(\frac{7 y}{2} \right)}} & \text{otherwise} \end{cases}\right) - 1$$
     //                   0                      for 7*y mod pi = 0\
     ||                                                            |
     ||                   2/7*y\                                   |
     ||              4*cos |---|                                   |
     ||                    \ 2 /                                   |
     ||----------------------------------------      otherwise     |
     ||                      2                                     |
-1 + |
            
$$\left(\begin{cases} 0 & \text{for}\: 7 y \bmod \pi = 0 \\\frac{4 \cos^{2}{\left(\frac{7 y}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{7 y}{2} \right)}}{\cos^{2}{\left(\frac{7 y}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \cos^{2}{\left(\frac{7 y}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) - 1$$
     //                 0                    for 7*y mod pi = 0\
     ||                                                        |
     ||                 2/7*y\                                 |
     ||            4*csc |---|                                 |
     ||                  \ 2 /                                 |
     ||------------------------------------      otherwise     |
     ||                    2                                   |
-1 + |
            
$$\left(\begin{cases} 0 & \text{for}\: 7 y \bmod \pi = 0 \\\frac{4 \csc^{2}{\left(\frac{7 y}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{7 y}{2} \right)}}{\csc^{2}{\left(- \frac{7 y}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2} \csc^{2}{\left(- \frac{7 y}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) - 1$$
-1 + Piecewise((0, Mod(7*y = pi, 0)), (4*csc(7*y/2)^2/((1 + csc(7*y/2)^2/csc(pi/2 - 7*y/2)^2)^2*csc(pi/2 - 7*y/2)^2), True))