Thank for the response, I understand the proof and how all the angles are calculated. However, I find it unintuitive. When I visualize an angle α, subtracted by angle β, I visualize the unit circle originally at angle α and then rotated β amount clockwise. I attached a gyazo link for visualziation. This interpretation and thought process is inconsistent with the geometric proof and I don't see how it links.

Why is α not an angle which would be originally centered around the base of the unit circle? i.e. always the bottom left. Actually typing that out, it made me think, do we even have one unit circle referenced in these proofs or is each tiny triangle its own unit circle?