<s>[INST] 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:
$$\left(1+\dfrac{1}{\sin a}\right)\left(1+\dfrac{1}{\cos a}\right)\ge 3+2\sqrt{2}; \text{ for } a\in\left]0,\frac{\pi}{2}\right[$$
 [/INST] Here is another proof:
Assume $x^2+y^2=1$ then $(x+y)^2=1+2xy$ and 
$$\left(1+\frac{1}{x}\right)\left(1+\frac{1}{y}\right)=\frac{xy+x+y+1}{xy}=\frac{(x+y+1)^2}{(x+y)^2-1}=\frac{x+y+1}{x+y-1}$$ 
Now rearranging the inequality becomes $x+y\leq \sqrt{2}$. And it can be seen simply that this is the maximum value of $x+y$ given $x^2+y^2=1$, for example $z=x+y$ is a plane and its intersection with the cylinder $x^2+y^2=1$has a maximum at $x=y$.
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