# Derivation of Margolus-Levitin bound on the Quantum Speed Limit

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?