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max |1 - a \cos x| >= \tan^2 (t/2) when |x|<= t, a>0, 0<t< \pi/2

Source: 1971 Swedish Mathematical Competition p5

March 26, 2021
algebratrigonometryinequalitiesmax

Problem Statement

Show that max⁡∣x∣≤t∣1−acos⁡x∣≥tan⁡2t2 \max\limits_{|x|\leq t} |1 - a \cos x| \geq \tan^2 \frac{t}{2} for aa positive and t∈(0,π2)t \in (0, \frac{\pi}{2}).
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