Find minimum value that the trigonometric expression may take

Question

For $x\in\left(0, \frac{\pi}{2}\right)$ find a minimal value, which the expression

$$\sec x+\csc x+\sec^{2}x+\csc^{2}x$$

can take.

My attempt:

I followed the trigonometrical approach and obtained

$$\sec x+\csc x+\sec^{2}x+\csc^{2}x=\sqrt{\left(2\csc 2x+1\right)^2-1}+4\csc^{2}2x\geq \sqrt{(2+1)^2-1}+4(1)=4+2\sqrt{2}$$

Above was obtained after lot of manipulations with the trigonometrical identities so I am looking for an easy approach to this problem.

Answer 1

Let $\sin{x}=a$ and $\cos{x}=b$.

Hence, $a^2+b^2=1$ and by AM-GM we obtain: $$\sec x+\csc x+\sec^{2}x+\csc^{2}x=$$ $$=\frac{a+b}{ab}+\frac{1}{a^2b^2}\geq\frac{2\sqrt2}{\sqrt{a^2+b^2}}+\frac{4}{(a^2+b^2)^2}=4+2\sqrt2.$$ The equality occurs for $a=b=\frac{1}{\sqrt2}$, which says that we got a minimal value.