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Comment by hn_throwaway_99 | Hacker News Reader

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.

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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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