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Trigonometric inequality

Source: Kvant Magazine No. 8 2022 M2712

March 8, 2023
Kvantgeometrytrigonometryinequalities

Problem Statement

Let ABCABC be a triangle, with ∠A=α,∠B=β\angle A=\alpha,\angle B=\beta and ∠C=γ\angle C=\gamma. Prove that ∑cyctan⁡α2tan⁡β2cot⁡γ2⩾3.\sum_{\text{cyc}}\tan \frac{\alpha}{2}\tan\frac{\beta}{2}\cot\frac{\gamma}{2}\geqslant\sqrt{3}.Proposed by R. Regimov (Azerbaijan)
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