Mathc initiation/a0024

Vérifier quelques propriétés mathématiques de trigonométrie

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

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

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

                     = 2 [  (1+cos(x))/2 (1+cos(y))/2                     
                          - (1-cos(x))/2 (1-cos(y))/2]                          
                          
                     = 2 [  (1+cos(x))  (1+cos(y)) /4                     
                          - (1-cos(x))  (1-cos(y)) /4]                            
                          
                     = 1/2 [  (1+cos(x))  (1+cos(y))                      
                            - (1-cos(x))  (1-cos(y)) ]     
                            
                           
                     = 1/2 [   1+cos(y)+cos(x)+cos(x)cos(y)                      
                            - (1-cos(y)-cos(x)+cos(x)cos(y)) ]                              
                                                  
     
                     = 1/2 [   1+cos(y)+cos(x)+cos(x)cos(y)                      
                            -  1+cos(y)+cos(x)-cos(x)cos(y) ]       
     
                      = 1/2 [  cos(y)+cos(x)                      
                              +cos(y)+cos(x) ]       
    
                      = 1/2 [  2cos(y)+2cos(x) ]       
     
                      =         cos(y) + cos(x)