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(edited for clarification) When I say "why," my question is not about people's individual reasons for thinking that there can't be any non-circular "trigonometric proof" or proof "based on trigonometry." What I mean is, how did it happen that so many people were thinking that way? Also, I'm not asking for people's opinions about it here. I'm asking, how can I find an answer to that question?(end edit)

Recently there was a lot of public discussion about proofs of the Pythagorean theorem "based on trigonometry." Without going into questions of what anyone might mean by "trigonometric proof" or proof "based on trigonometry," I've been wondering how widespread was the idea that any such proof would have to be circular, and in particular if there were any people with math degrees who were saying that. I've seen people saying that in comments on Youtube and Twitter, and one math blogger saying that he and many others were thinking that at one time. I have an idea about why people were thinking that. I'm especially curious to know if there were any people with math degrees who were thinking that. All I can think of to do is just choose some people with math degrees randomly in an Internet search, and ask them about it. Another idea is that maybe I could find someone who would take it on as a research project. Does anyone have any better idea how I can find out how widespread that idea was, and if there were any people with math degrees who were thinking that way?

(later) I think I'll just find a few people with math degrees to talk to about it, asking if they or any mathematicians they knew ever thought that there was anything revolutionary about a "trigonometric proof" or a proof "based on trigonometry." Then if I find any people with math degrees who were thinking that way, try to find out why.

Lamont
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    I don't think it means all that much. On its face, it seems tricky to work with the trig functions without using Pythagorus. Identities like $\sin^2 \theta +\cos^2\theta =1$ are ubiquitous. Thus, if you are using triangle ratios in any guise, you must take great care to avoid circularity. I doubt people thought about it much more than that. – lulu May 05 '23 at 14:23
  • It was a limitation in people's thinking that I think might be a result of teaching practices and how textbooks are written. I'm thinking that maybe I'll just search for people with math degrees to talk to about it, and maybe that will give me some ideas for a better way to find answers to my questions about it: How widespread was the idea that it was impossible for there to be any valid "trigonometric proof" or proof "based on trigonometry," why were people thinking that, and how many of them were people with math degrees? – Lamont May 05 '23 at 14:45
  • Don't forget that the Pythagorean Theorem is equivalent to the Parallel Postulate, so you can't achieve a proof (of any sort) without some assumption equivalent to the Parallel postulate. Thus the argument must fail for, say, spherical geometries or hyperbolic geometries. I have not reviewed the high school proof, but it must have used some assumption equivalent to the Parallel Postulate (maybe they used that the angles in a triangle sum to $\pi$? that's another equivalent). – lulu May 05 '23 at 15:00
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    Ah, I see they used similarity. One axiom equivalent to the Parallel Postulate is that there exists a pair of similar triangles which are not congruent. – lulu May 05 '23 at 15:04
  • The reason for such thinking is obvious: the most fundamental relation in trigonometry is $\sin^2\theta + \cos^2\theta = 1$, which is basically equivalent to the Pythagorean theorem. So it is reasonable to think it is unlikely to prove that theorem via trigonometry in a non-circular way (pun intended). Nobody would claim by pure logic that such a proof is impossible, but simply unlikely. That there in fact turn out to be such proofs is not revolutionary: that label is too strong. It is simply interesting, but such proofs are not going to change math in a dramatic way. – KCd May 06 '23 at 02:55
  • This may help: https://math.stackexchange.com/questions/3566541/is-pythagoras-theorem-a-theorem/3566726#3566726 – Ethan Bolker May 06 '23 at 03:01

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