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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?

Thank you in advance.

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    $\begingroup$ That's one of the most usual methods to solve identities: develop mathematically one of the sides until you an expression equal to the other side. Alas, this method is not always sufficient. $\endgroup$ – DonAntonio Jan 19 at 21:44
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    $\begingroup$ As for your third question, it relies on the assumption that ALL trig problems will be able to be written as elementary trig functions. Even if this was the case, sometimes its way easier just to use a proof by contradiction $\endgroup$ – Aniruddh Venkatesan Jan 19 at 21:50

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