There is a proof of the addition formula for cosines cos(a-b) that uses a unit circle and the law of cosines to arrive at the law. My question is does this proof only work on the unit circle? In the proof, you substitute 1^2 for two sides of the triangle because they are radii and the radius is 1 unit long in this case. Later in the proof you use sin^2 + cos^2 =1 and everything works out fine. But what if the radius was 2? Wouldn't the proof fail?