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In their derivation of the Margolus-Levitin bound on the "quantum speed limit" the authors make use of a trigonometric inequality

$\cos x \ge 1-\frac{2}{\pi}(x+\sin x), \quad \forall x\ge0$

Is there an intuitive reason why this inequality is used? It seems such an arbitrary choice. Failing that, how do you prove this inequality?

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