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sum sin (4n-3)a_n/ (1-cos (4n-3)a_n) =kn

Source: IV Soros Olympiad 1997-98 Corr. 11.10 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

June 1, 2024
algebratrigonometrySum

Problem Statement

Let an=π2na_n = \frac{\pi}{2n}, where nn is a natural number. Prove that for any k=1k = 1,22,......, nn holds the equality sin⁡kan1−cos⁡an+sin⁡5kan1−cos⁡5an+sin⁡9kan1−cos⁡9an+...+sin⁡(4n−3)an1−cos⁡(4n−3)an=kn\frac{\sin ka_n}{1-\cos a_n}+\frac{\sin 5ka_n}{1-\cos 5a_n}+\frac{\sin 9ka_n}{1-\cos 9a_n}+...+\frac{\sin (4n-3)a_n}{1-\cos (4n-3)a_n}=kn
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