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

Answer: 
By the AM-GM inequality, $$\dfrac{1}{\cos a}+\dfrac{1}{\sin a} \geq 2 \sqrt{\dfrac{1}{\sin a\cos a}}$$

Since $\sin a \cos a = \frac12 \sin(2a) \leq \frac12$, we have $\dfrac{1}{\sin a\cos a}\geq 2$.

Hence $$\left(1+\dfrac{1}{\sin a}\right)\left(1+\dfrac{1}{\cos a}\right) = 1+\dfrac{1}{\cos a}+\dfrac{1}{\sin a}+\dfrac{1}{\sin a\cos a} \\\geq 1+2\sqrt{\dfrac{1}{\sin a\cos a}}+\dfrac{1}{\sin a\cos a}\geq 1+2\sqrt{2} + 2 = 3+ 2\sqrt{2}$$
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 $a = \frac{\pi}{4}$