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Comment by jonahx | Hacker News Reader

> What am I missing?

Nearly all the nontrivial results of trigonometry do in fact rest on the pythagorean theorem. The trig identities you learned in high school, as well as more advanced results like power series, etc. These results would be inadmissible.

So the “uses trigonometry” part of this story feels like an attempt to manufacture mystery and hype. Which is a shame, because the geometric series construction is imo the interesting part, and can stand on its own merits.

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Apparently (found this while reading an article on the girls' accomplishments) somebody proved sin^2x + cos^2x = 1 without using the Pythagorean theorem in 2009: https://forumgeom.fau.edu/FG2009volume9/FG200925index.html.

I don't think the "uses trigonometry" part is hype. They do use the definitions and law of sines, they just cleverly avoid the parts of trigonometry that depend on the Pythagorean theorem.

I address this in my OP. If you watch the video of the proof, you will see that the "law of sines" is 1 step away from the ratio definition of sin. You just drop one altitude, apply the definition again to the similar triangles, and re-arrange. It is almost content free as a result -- I see no reason using this in a proof would have special significance. For example, the standard proof using similar triangles (https://sumantmath.wordpress.com/2020/08/16/proof-of-pythago...) is implicitly using the law of sines.

The hype part is the implication that impossible trig barrier was shattered by their proof.

Not at all - the hype part is that 2 teenagers derived a uniquely elegant proof that other highly-trained mathematicians had thus far failed to do so.

Any other claims seem to have been added by the media, not the teenagers themselves.

That is what I was referring to. I wasn’t accusing the teenagers of anything.

This theorem is essentially the Pythagorean theorem, so … a proof of one is a proof of the other.

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