I was tasked with proving a trig identity -- the particular identity doesn't much matter, as I'm interested far more in whether this is a valid method of proof, generally -- and a professor recommended this strategy: equate both sides and, via dot and cross products and rules of vector algebra, produce a trivially true statement. The conclusion, then, is that the original identity is correct.
This strategy is a bit new to me. I've usually been told to start on one side of the equation and generate the other. But, in thinking about this more, and specifically in trying to provide a counterexample, it seems that this strategy is valid provided that we can reverse each of the steps we took to get down to some trivial statement. In various proofs involving inequalities, for example, which require reversing signs, this is likely to fail.
Is my intuition here correct? I would be very interested in hearing of possible counterexamples to this or conditions where a proof like this is appropriate, as well as whether such a proof is frowned upon -- proofs like this do seem rather uncommon, and I believe this may be the first time I've seen such a statement proved like this.