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Ugly trigonometric inequality [Nordic 2017, P2]

Source: Nordic Mathematical Contest 2017 Problem 2

April 4, 2017
trigonometryinequalities

Problem Statement

Let a,b,α,βa, b, \alpha, \beta be real numbers such that 0≤a,b≤10 \leq a, b \leq 1, and 0≤α,β≤π20 \leq \alpha, \beta \leq \frac{\pi}{2}. Show that if abcos⁡(α−β)≤(1−a2)(1−b2), ab\cos(\alpha - \beta) \leq \sqrt{(1-a^2)(1-b^2)}, then acos⁡α+bsin⁡β≤1+absin⁡(β−α). a\cos\alpha + b\sin\beta \leq 1 + ab\sin(\beta - \alpha).
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