Consider a regular
![2n](/media/m/d/2/d/d2da874dc9bc356be9468cdbd57fbfdf.png)
-gon and the
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
diagonals of it that pass through its center. Let
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
be a point of the inscribed circle and let
![a_1, a_2, \ldots , a_n](/media/m/0/a/8/0a84730daafb8c167c30263462061224.png)
be the angles in which the diagonals mentioned are visible from the point
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
. Prove that
%V0
Consider a regular $2n$-gon and the $n$ diagonals of it that pass through its center. Let $P$ be a point of the inscribed circle and let $a_1, a_2, \ldots , a_n$ be the angles in which the diagonals mentioned are visible from the point $P$. Prove that
$$\sum_{i=1}^n \tan^2 a_i = 2n \frac{\cos^2 \frac{\pi}{2n}}{\sin^4 \frac{\pi}{2n}}.$$