Proof of Compound Angle from Ptolemy's Theorem

Question

I have a query regarding a proof I'm reading on the additive Sine compound angle formula, which uses Ptolemy's theorem.

http://www.cut-the-knot.org/proofs/sine_cosine.shtml

I'm looking at the additive Sine section. The line BC is defined to be the diameter of the circle and is of length 1. All the sides except AD are obvious through basic trigonometry as the angle BAC=BDC=90 degrees. The final line is the application of Ptolemy's theorem.

However, I do not understand how they have arrived at the fact $AD = sin(\alpha + \beta)$. Can someone shed some light on how this is the case?

Answer 1

By just reading through the proof:

"The Law of Sines supplies the length of the remaining diagonal."

So just do that: go back to formula $(3)$ in the paper and note that

$$\frac{\sin(\alpha+\beta)}{AD}=2R=BC=1,$$

which is exactly the statement you were looking for