Mathc initiation/a0025

Vérifier quelques propriétés mathématiques de trigonométrie

Vérifions si :  sin(x) - sin(y) = 2 sin( (x-y)/2 ) cos( (x+y)/2 )

  posons :
     
     sin(x) - sin(y) = 2 sin( (x-y)/2 )   cos( (x+y)/2 )    
         
  Soit :  
                     = 2 sin( x/2 - y/2 ) cos( x/2 + y/2 )    
                                           
 
     Nous avons vu que :
    
     sin(x-y) = sin(x) cos(y) - cos(x) sin(y) 
     cos(x+y) = cos(x) cos(y) - sin(x) sin(y)   
     
     
     
   donc 

     sin(x) - sin(y) = 2 [sin( x/2 - y/2 )]
                         [cos( x/2 + y/2 )]
     
                     = 2 [sin(x/2) cos(y/2) - cos(x/2) sin(y/2)]
                         [cos(x/2) cos(y/2) - sin(x/2) sin(y/2)]
                         
                     = 2 [+ cos(y/2)**2 sin(x/2)cos(x/2)
                          - sin(x/2)**2 sin(y/2)cos(y/2)
                                 
                          - cos(x/2)**2 sin(y/2)cos(y/2)
                          + sin(y/2)**2 sin(x/2)cos(x/2)]                        
                         
                     
                     
                     = 2 [- sin(x/2)**2 sin(y/2)cos(y/2)                     
                          - cos(x/2)**2 sin(y/2)cos(y/2)
                          
                          + cos(y/2)**2 sin(x/2)cos(x/2)
                          + sin(y/2)**2 sin(x/2)cos(x/2)]    
                          
                     
                     = 2 [+ cos(y/2)**2 sin(x/2)cos(x/2)
                          + sin(y/2)**2 sin(x/2)cos(x/2)
                          
                          - sin(x/2)**2 sin(y/2)cos(y/2)                     
                          - cos(x/2)**2 sin(y/2)cos(y/2)]                             
                          
                             
                     = 2 [+ cos(y/2)**2 sin(x/2)cos(x/2)
                          + sin(y/2)**2 sin(x/2)cos(x/2)
                                      -
                        (   sin(x/2)**2 sin(y/2)cos(y/2)                     
                          + cos(x/2)**2 sin(y/2)cos(y/2) )]                            
                             
                                                    
                                                                       cos(x)**2+sin(x)**2=1 
                         
                     = 2 [ (cos(y/2)**2+sin(y/2)**2) sin(x/2)cos(x/2)
                                      -
                           (sin(x/2)**2+cos(x/2)**2) sin(y/2)cos(y/2) ]                            
                             
                         
                         
                     =   2[ sin(x/2)cos(x/2) - sin(y/2)cos(y/2) ]   
                     
                                                                        sin(x/2) = sqrt((1-cos(x))/2)
                                                                        cos(x/2) = sqrt((1+cos(x))/2)
                                                                        
                    =    2[ sqrt((1-cos(x))/2) sqrt((1+cos(x))/2) - 
                            sqrt((1-cos(y))/2) sqrt((1+cos(y))/2) ]
   

                                                                          sqrt(x) sqrt(y) = sqrt(xy)
                     =   2[ sqrt((1-cos(x))/2)  (1+cos(x))/2) -
                            sqrt((1-cos(y))/2)  (1+cos(y))/2)  ]
                                                                 
                                                                 
                     =   2[ sqrt((1-cos(x))  (1+cos(x)) /4) -
                            sqrt((1-cos(y))  (1+cos(y)) /4)  ]
                                                                 
                                                                 
                     =      sqrt(  (1-cos(x))  (1+cos(x)) ) -             (x-y)(x+y) = x**2-y**2
                            sqrt(  (1-cos(y))  (1+cos(y)) )   
                            
          
                     =      sqrt(  (1-cos(x)**2)  ) -                    sin(x)**2+cos(x)**2=1  
                            sqrt(  (1-cos(y)**2)  )                      
                                                                         1-cos(x)**2 = sin(x)**2 
                                                                          
                     =      sqrt(sin(x)**2) -                             
                            sqrt(sin(y)**2)                               
                                 
                      =      sin(x) - sin(y)