Proving trigonometric identities means to show that one side is equal to the other.
You can do it in three ways:
a. Start with the LHS and show that it equals to the RHS.
b. Start with the RHS and show that it equals to the LHS.
c. Work with both sides simultaneously until you arrive at the same expression for both.
WARNING: Some teachers like myself love to give exercises where some identities are false. So how can you avoid wasting your time trying to prove such identities true, when in fact they are false and can't be proved?
You try substituting an obscure angle (say, #17.314^\circ#) and checking if both sides are equal. If they are not, then don't waste your time because you have found a counterexample . Stop and write: Can't be proved because it is false.