The maximum value of $|\sqrt{{\sin }^{2}x+2a^2}-\sqrt{2a^2-1-{\cos }^{2}x}|,$ where a, $x\in R$ is

For maximum value
$2a^2-1-{\cos }^{2}x=0\Rightarrow {\cos }^{2}x=2a^2-1$ 
$\Rightarrow 1-{\sin }^{2}x=2a^2-1\Rightarrow {\sin }^{2}x=2-2a^2$ 
${\sin }^{2}x+2a^2=2$ 
$\therefore$ Maximum value of $|\sqrt{{\sin }^{2}x+2a^2}-\sqrt{2a^2-1-{\cos }^{2}x}|=|\sqrt{2}-0|=\sqrt{2}$