ILL 1982 Problem 13

A regular $n$ -gonal truncated pyramid is circumscribed around a sphere. Denote the areas of the base and the lateral surfaces of the pyramid by $S_1, S_2$ , and $S$ , respectively. Let $\sigma$ be the area of the polygon whose vertices are the tangential points of the sphere and the lateral faces of the pyramid. Prove that $$\sigma S = 4S_1S_2 \cos^2 \frac{\pi}{n}.$$