The maximum value of expression $ \sqrt{\sin^2x+ 2a^2} - \sqrt{-1 -\cos^2x+ 2a^2} $ If $a,x\in\Bbb R$, what is the maximum value of the expression 
$ \sqrt{\sin^2x+ 2a^2} - \sqrt{-1 -\cos^2x+ 2a^2} $?
I tried to differentiate but it became messy.

Let $$ f(x) = \sqrt{\sin^2x+ 2a^2} - \sqrt{2a^2-1-\cos^2 x} $$
Now for $\max$ of $f(x)\;,$ Put $$2a^2-1-\cos^2 x=0\Rightarrow 2a^2=1+\cos^2 x$$
So $$f_{\max} = \sqrt{\sin^2 x+1+\cos^2 x} = \sqrt{2}$$
