How to prove $\left(1+\frac{1}{\sin a}\right)\left(1+\frac{1}{\cos a}\right)\ge 3+2\sqrt{2}$? Prove the inequality:
$$\left(1+\dfrac{1}{\sin a}\right)\left(1+\dfrac{1}{\cos a}\right)\ge 3+2\sqrt{2}; \text{ for } a\in\left]0,\frac{\pi}{2}\right[$$

I think it should be $(0,\frac{\pi}{2})$, so that both $\sin a$ and $\cos a$ are positive and well defined.
$$\bigg (1+\frac{1}{\sin a} \bigg)\bigg (1+\frac{1}{\cos a} \bigg) = 1+ \frac{1}{\sin a} + \frac{1}{\cos a} + \frac{1}{\sin a\cos a}$$
$$\frac{1}{\sin a} + \frac{1}{\cos a} \geq \frac{4}{\sin a + \cos a} \geq \frac{4}{\sqrt{2(\sin^2 a + \cos^2 a)}} = 2\sqrt 2$$
$$\frac{1}{\sin a\cos a} \geq \frac{2}{\sin^2 a + \cos^2 a} = 2$$
So,
$$\bigg (1+\frac{1}{\sin a} \bigg)\bigg (1+\frac{1}{\cos a} \bigg) = 1+ \frac{1}{\sin a} + \frac{1}{\cos a} + \frac{1}{\sin a\cos a} \geq 3+ 2\sqrt 2$$
Equality occurs when $a = \frac{\pi}{4}$
