Mathc initiation/a007

Vérifier quelques propriétés mathématiques de trigonométrie

                           3tan(x) -tan(x)**3
Vérifions si :  tan(3x ) = ------------------
                              1-3tan(x)**2

    Nous avons vu que :
    
                             tan(y) +  tan(x)
                 tan(y+x)  = ----------------     
                             1 - tan(y)tan(x)
                                                       
     
    posons y = 2x :     
    
                             tan(2x) +  tan(x)    
                 tan(2x+x) = ----------------     
                             1 - tan(2x)tan(x)     
                             

                             tan(2x) +  tan(x)    (a)
                  tan(3x)  = ----------------     
                             1 - tan(2x)tan(x)    (b)
                              
 a) ------------------------------------------
 
                          2tan(x)
    tan(2x) +  tan(x) = ----------- +  tan(x)                        
                        1-tan(x)**2
                          
                          
                          2tan(x)      tan(x)  (1-tan(x)**2)
                      = ----------- +  ---------------------                        
                        1-tan(x)**2       1-tan(x)**2
                        
                                
                          2tan(x)      tan(x)-tan(x)**3
                      = ----------- +  ----------------                       
                        1-tan(x)**2      1-tan(x)**2
                                      
                                      
                          3tan(x) -tan(x)**3
                      = --------------------                      
                            1-tan(x)**2 
                            
                            
  b) ------------------------------------------       
  
                             2tan(x)
  1 - tan(2x)tan(x) =  1 - -----------  tan(x)
                           1-tan(x)**2
  
                       1-tan(x)**2   2tan(x)**2
                    =  ----------- - ----------- 
                       1-tan(x)**2   1-tan(x)**2    
                           
                       1-3tan(x)**2   
                    =  ------------     
                        1-tan(x)**2        
                           
                           
 a/b) ------------------------------------------                            
                           
                           
                          3tan(x) -tan(x)**3
                          ------------------                      
                            1-tan(x)**2                            
            tan(3x ) = ------------------------  
                            1-3tan(x)**2   
                            ------------     
                            1-tan(x)**2                                
                           
  donc                          
                           
                       3tan(x) -tan(x)**3    
            tan(3x ) = ------------------                         
                          1-3tan(x)**2