# Is simplifying a complete expansion of the right hand side of a trigonometric identity sufficient to prove it?

My first question is, when proving a trigonometric identity for all real values of a variable such as $$x$$, is it sufficient to expand the right hand side into elementary functions and simplify until it equals the left hand side?

I know this is very basic but since I have seen complicated proofs of identities by induction, I want to be sure that this is a sufficient method of proof if the problem allows. My second question is, is this considered a direct proof?

My third question is, assuming that the answers to my first two questions are affirmative, would there be any reason that a trigonometric identity need be proven any way other than the way I specified in my first question? If trigonometric expressions can be rewritten in terms of the elementary trig functions, would there ever be need for an induction proof rather than direct?