BG-MO 1971 Problem 5

Let $A_1,A_2,\ldots,A_{2n}$ are the vertices of a regular $2n$ -gon and $P$ is a point from the incircle of the polygon. If $\alpha_i=\angle A_iPA_{i+n}$ , $i=1,2,\ldots,n$ . Prove the equality $$\sum_{i=1}^n\tan^2\alpha_i=2n\frac{\cos^2\frac\pi{2n}}{\sin^4\frac\pi{2n}}.$$