Derivation of the Sum-to-Product Formulas

 

     To prove the sum-to-product formulas, we shall use the product-to-sum formulas. They are included below, labeled (1) through (4).

 

(1)  

(2)  

(3)  

(4)  

 

     Before we begin, though, let us make a change of variables. Let us set  and . We see that  and .

 

     Using these substitutions, we can now derive the sum-to-product formulas.

 

Formula (1) becomes:

 

      

 

Formula (2) becomes:

 

      

 

Formula (3) becomes:

 

      

 

Formula (4) becomes: