https://mathoverflow.net/questions/71691/geometric-meaning-of-a-trigonometric-identity
In the question on mathoverflow that is linked above, I mentioned that I had proved an identity by mathematical induction. Noam Elkies, a professor at Harvard, posted an answer. He did not answer the question actually posed. Rather, he posted a better proof of the identity that I had proved by induction.
Is his proof actually better? My inclination is to say "yes", but I'm not staking my life on that.
But some years ago, I concluded that when a proposition can be proved either by mathematical induction or by other methods, the proof by other methods is usually better. This was based in part on various particular examples. But I can't remember what any of those are!
So was I right? And if so, what are (1) the examples (hundreds of them, if you have them!), and (2) the explication of how they are better?