Mathc initiation/a0049

Vérifier quelques propriétés mathématiques de trigonométrie hyperbolique

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

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

                               tanh(2x) + tanh(x)    (a)
                  tanh(3x)  =  --------------------
                               1 + tanh(2x) tanh(x)  (b)                           


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