Derivation of the Product-to-Sum Formulas

 

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

 

     (1)  

     (2)  

     (3)  

     (4)  

 

 

If we add formula (1) and formula (2), we have:

 

      

 

If we subtract formula (2) from formula (1), we have:

 

      

 

If we add formula (3) and formula (4), we have:

 

      

 

If we subtract formula (3) from formula (4), we have: