This picture shows an animation of the geometry necessary to derive the cosine of a difference formula.

The sketch on the left shows the original geometry, with the angles u and v defining points A and B on the unit circle. The angle between the two terminal sides is θ = u - v. We can rotate the angle θ so that it is in standard position. This gives us two new points on the unit circle C and D.

From the diagram, we see that the distance between the points A and B is the same as the distance between the points C and D. Using the distance between two points formula, we can derive the formula for the cosine of a difference:

cos (u-v) = cos u cos v + sin u sin v.