5.4  Sum and Product Identities

It is occasionally advantageous to rewrite sums of sines and cosines as products or to rewrite products as sums. For example, it might be easier to tell when a product is equal to zero by setting each factor to zero than it is to tell when a sum of terms is zero. The sum and difference formulas are used to create new identities to do this type of conversion. First we state the most common identities in this table.

Product to Sum Formulas Sum to Product Formulas

All eight of these formulas are easily obtained from sum and difference identities. Here is a demonstration of the first one just to see how they are created.

A Formula Derivation

Here is one example of how the first Sum to Product formula could be used to solve an equation:

Example: Solve the equation sin(4x) + sin(2x) = 0.
Soltion: By the formula this is equivalent to 2sin(2x)cos(x) = 0. It's fairly easy then to see that x = 0 is a solution to sin(2x) = 0 and x = /2 is a solution to cos(x) = 0. In fact all solutions can be obtained by just setting each factor to zero.