The wikipedia provides geometrical interpretations for the angle addition identities, but not for the power reducing identities. My students and I have been playing around with them and arrived at a geometric interpretation of the power reduction for $\sin^2\theta$ that we've never seen before. What would be a good place to search to find out whether our geometric interpretation is novel?
Essentially the argument goes like this:
Draw two congruent right triangles such that their right angles are coincident, their smaller angle $\theta$ and their hypotenusae $1$. Construct a square, $A$ whose side length is $\sin\theta$ and another square $B$ whose side length is $\cos\theta$. The gnomon inside $B$ but not inside $A$ has area $\cos2\theta$ by the angle addition identity. Draw a square $C$ of side length $1$ circumscribing $B$. The area inside $C$ but not inside $B$ has area $\sin^2\theta$ by the Pythagorean identity. Thus, $\sin^2\theta + \sin^2\theta + \cos2\theta$ must equal $1$.
Is that a well known argument?