Prove that $$\frac{\sin^3 a}{\sin b} +\frac{\cos^3 a}{\cos b} \ge \frac{1}{\cos(a - b)}$$for all $a$ and $b$ in the interval $(0, \pi/2)$ .
Source: 2011 Saudi Arabia Pre-TST November 3.1
Tags: algebra, inequalities, trigonometry
Prove that $$\frac{\sin^3 a}{\sin b} +\frac{\cos^3 a}{\cos b} \ge \frac{1}{\cos(a - b)}$$for all $a$ and $b$ in the interval $(0, \pi/2)$ .