Mathc initiation/a0035

Vérifier quelques propriétés mathématiques de trigonométrie hyperbolique

Vérifions si :  cosh(x)**2 - sinh(x)**2 = 1

a)           *******************
 
                sinh(x) =  (e**x - e**(-x)) / 2
                cosh(x) =  (e**x + e**(-x)) / 2
                
  
              2 sinh(x) =   e**x - e**(-x)    
              2 cosh(x) =   e**x + e**(-x)   
              
              
              2 sinh(x) =   e**x - e**(-x)   
            + 2 cosh(x) =   e**x + e**(-x)               
              ----------------------------
  2 sinh(x) + 2 cosh(x) =   e**x - e**(-x) + e**x + e**(-x)     
      
              
  2 sinh(x) + 2 cosh(x) =   e**x           + e**x             
              
  2 sinh(x) + 2 cosh(x) = 2 e**x                        
              
    sinh(x) +   cosh(x) =   e**x               
              
              
           e**x = sinh(x) +   cosh(x) 
              
                

 b)           *******************
 
                sinh(x) =   (e**x - e**(-x)) / 2
                cosh(x) =   (e**x + e**(-x)) / 2
                
  
              2 sinh(x) =   e**x - e**(-x)    
              2 cosh(x) =   e**x + e**(-x)   
              
              
              2 sinh(x) =   e**x - e**(-x)   
            - 2 cosh(x) =   e**x + e**(-x)               
              ----------------------------
  2 sinh(x) - 2 cosh(x) =   e**x - e**(-x) - (e**x + e**(-x))     
  
  2 sinh(x) - 2 cosh(x) =   e**x - e**(-x) - e**x  - e**(-x)  
  
  2 sinh(x) - 2 cosh(x) =        - e**(-x)         - e**(-x)   
  
  2 sinh(x) - 2 cosh(x) =      - 2 e**(-x)         
  
    sinh(x) -   cosh(x) =      -   e**(-x)   
  
  
              -   e**(-x)  =   sinh(x) - cosh(x)      
  
                  e**(-x)  =   cosh(x) - sinh(x)  
  
  

 c)           *******************
 
   
               e**x     =  sinh(x) + cosh(x)           a)
               e**(-x)  =  cosh(x) - sinh(x)           b)
  
  
               e**x     =  cosh(x) + sinh(x)  
          *    e**(-x)  =  cosh(x) - sinh(x)  
               -----------------------------
        e**x  e**(-x)   =  (cosh(x) + sinh(x))  (cosh(x) - sinh(x))        
        
                                                                    (a+b)(a-b) = a**2-b**2
             e**(x-x)   =     cosh(x)**2 - sinh(x)**2    
             
               e**(0)   =     cosh(x)**2 - sinh(x)**2       
               
                  1     =     cosh(x)**2 - sinh(x)**2