(5-34)            ,

(5-35)            .  

Proof.

    Subtracting Equation (5-31) from Equation (5-30) and solving for gives

               

where    and  ,  respectively, are the hyperbolic cosine and hyperbolic sine functions that you studied in calculus.

    Similarly        

(c) 2006 John H. Mathews, Russell W. Howell