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so were going to see how the angle sum and difference formula for sine can rise naturally as a consequence of ptolemys theorem the autonomys theorem tells you that if youve got a cyclic quadratural ABCD then the side lengths and also the lengths of the diagonals AC and BD satisfy this equation so with the correct geometric setup we can actually quite easily show that the angle sum formula for sine holds so the setup that we need is that first of all AC goes through the center so this is a diameter of the circle and were also going to choose this so that length of the diameter is one so weve got our angles Theta and Phi that were interested in here so were going to try and prove the sine theta plus Phi identity and weve also got this set up so that this the diameter which means that the angle at B here is going to be a right angle and similarly your angle down at D here is also going to be a right angle so this turns out to be really useful because now youre triangle ABC is a
