How to Apply the Sine Sum and Difference Formulas to Trig Proofs
When dealing with sine sum and difference formulas in a trig proof, you need to make one side of the given equation look like the other. You can work on both sides to get a little further if need be, but make sure you know how your teacher wants the proof to look.
For example, when asked to prove
follow these steps:

Look for identities in the equation.
In this example, you can see the sum identity,
and the difference identity,
for sine.

Substitute for the identities.

Simplify to get the proof.
Two terms cancel, leaving you with this equation:
Combine the like terms to get the answer: