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: