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A regular 2n-gon and the n diagonals - Prove the identity

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September 23, 2010
trigonometrypolygonTrigonometric IdentitiesgeometryIMO Shortlist

Problem Statement

Consider a regular 2n2n-gon and the nn diagonals of it that pass through its center. Let PP be a point of the inscribed circle and let a1,a2,…,ana_1, a_2, \ldots , a_n be the angles in which the diagonals mentioned are visible from the point PP. Prove that ∑i=1ntan⁡2ai=2ncos⁡2π2nsin⁡4π2n.\sum_{i=1}^n \tan^2 a_i = 2n \frac{\cos^2 \frac{\pi}{2n}}{\sin^4 \frac{\pi}{2n}}.
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