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product sin \frac{k'\pi}{2n} (I Soros Olympiad 1994-95 Ukraine R2 10.5)

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June 6, 2024
trigonometryalgebraProduct

Problem Statement

For an arbitrary natural nn, prove the equality sin⁡π2nsin⁡3π2nsin⁡5π2n...sin⁡n′π2n=21−n2\sin \frac{\pi}{2n}\sin \frac{3\pi}{2n}\sin \frac{5\pi}{2n}...\sin \frac{n'\pi}{2n}=2^{\dfrac{1-n}{2}} where n′n' is the largest odd number not exceeding nn.
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