Nice. My favorite method for remembering those equalites is by using e^i(x+y)=e^ix * e^iy (Its not a proof unless you know that initial equation holds without using the geometric identities, but there are definitions of cos and sin demanding it)

@DrBarker

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Yes, the proofs using e^{i (x+y)} are definitely the cleanest! This one isn't too bad if you take Ptolemy's theorem and also the extended sine rule as given - this saves us the effort of finding the length of BD.

Very nice indeed & very unexpected application of Ptolemy's Theorem! I think we should also consider the case where O lies on BD. In this case θ+φ=90°, so sin(θ+φ)=1 and BD=1 (diameter) so we're good to go in this case also. BTW, your result that sin α=opposite side for circumscribed circle of diameter 1 is a particular case of the well known result that for a ΔABC, a/sin A=b/sin B=c/sin C=2R where R is the radius of the circumscribed circle. Moreover, your proof of this particular case becomes the proof of the general case if you take the radius of the circumscribed circle to be R instead of ½.

@DrBarker

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That's a good point about the case where O lies on BD. I suppose it is technically covered by the proof for either of the cases considered, albeit in a degenerate way! Yes, this is the extended sine rule in disguise - I didn't think this was well-known enough to just quote without at least a proof for the case here with r = 1/2.

@MichaelRothwell1

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@@DrBarker I agree that the extended sine rule is not as well known as it should be, which is a pity.

Golden, very helpful

So neat!

amazing

Strange to think that old Ptolemy could've come up with our modern trig functions with just a little more work. Maybe there was a good reason to avoid them though

Nice!