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Sum of tangent products

Source: Czech and Slovak Olympiad 1971, National Round, Problem 4

July 9, 2024
algebratrigonometrytangent

Problem Statement

Show that there are real numbers A,BA,B such that the identity ∑k=1ntan⁡(k)tan⁡(k−1)=Atan⁡(n)+Bn\sum_{k=1}^n\tan(k)\tan(k-1)=A\tan(n)+Bn holds for every positive integer n.n.
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