Maximum value

[jacks]

Maximum value of $f(x) = \left|\sqrt{\sin^2 x+2a^2}-\sqrt{2a^2-1-\cos^2 x}\right|\;,$ Where $a,x\in \mathbb{R}$

I have Tried like this way:: Here $f(x) = \left|\sqrt{\sin^2 x+2a^2}-\sqrt{2a^2+\sin^2 x-2}\right|$

Let $\sin^2 x+2a^2=y\;,$ Then $f(y) = \left|\sqrt{y}-\sqrt{y-2}\right| = \frac{2}{\left|\sqrt{y}+\sqrt{y-2}\right|}$

So for $y\geq 2\;,$ We get $\sqrt{y}+\sqrt{y-2}\geq \sqrt{2}$. So $\displaystyle \frac{1}{\sqrt{y}+\sqrt{y-2}}\leq \frac{1}{\sqrt{2}}$

So $\displaystyle f(y) = \frac{2}{\left|\sqrt{y}+\sqrt{y-2}\right|}\leq \frac{2}{\sqrt{2}}=\sqrt{2}$

But above process is very lengthy, How can we solve it using Inequality Directly.

Help me , Thanks

[Dan B-Yallay]

Is this problem about real- or complex variables? Because in the former case the parameter $a$ ought to be bigger or equal to 1 by absolute value.