sudgylacmoe | Deriving Every Trig Identity in a Minute @sudgylacmoe | Uploaded May 2023 | Updated October 2024, 4 hours ago.
In this short, I derive many of the most important trigonometric identities, starting from the Taylor series of the sine and the cosine. Credit for the sum identity of the sine comes from Michael Hoppe from math.stackexchange, who in turn got it from E. Schmidt. math.stackexchange.com/questions/635077/proof-of-the-angle-sum-identity-for-sin/635552#635552

I don't know how much people read the description of shorts, but there's actually a long story behind this one. I have a Coq project where I am deriving a bunch of math from the ground up (which you can find at github.com/sudgy/math-from-nothing ). Eventually I want to describe trigonometric functions (although I'm not there yet), and I've always wondered what the best way to prove the trig identities from the power series is. Because of this, I've always been on the lookout for sneaky and fast proofs of these identities, and I've collected all of the proofs that I've seen like this in this short.

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