NMC 2017 Problem 2

Let $a, b, \alpha, \beta$ be real numbers such that $0 \leq a, b \leq 1$, and $0 \leq \alpha, \beta \leq \frac{\pi}{2}$. Show that if

$$a b \cos (\alpha-\beta) \leq \sqrt{\left(1-a^{2}\right)\left(1-b^{2}\right)}$$

then

$$a \cos \alpha+b \sin \beta \leq 1+a b \sin (\beta-\alpha)$$