<s>[INST] Maximum value of $f(x) = \left|\sqrt{\sin^2 x+2a^2} - \sqrt{2a^2-1-\cos^2 x}\right|\;\;,$ Where $a,x\in \mathbb{R}$ Calculation of Maximum value of $\displaystyle f(x) = \left|\sqrt{\sin^2 x+2a^2} - \sqrt{2a^2-1-\cos^2 x}\right|\;\;,$ Where $a,x\in \mathbb{R}$
$\bf{My\; Try::}$ We Can Write It as $f(x) = \left|\sqrt{2a^2-\cos^2 x+1}-\sqrt{2a^2-\cos^2 x-1}\right|$
Now Let $\left(2a^2-\cos^2 x\right) = b\in \mathbb{R}\;,$ Then Expression is $f(b) = \left|\sqrt{b+1}-\sqrt{b-1}\right|\;,$ where $b\in \mathbb{R}.$
Now How Can I find max. value of $f(b)$
Help me
Thanks 
 [/INST] Hint: Let $g(b)=\sqrt{b+1}-\sqrt{b-1}$ where $b\in [2a^2-1,2a^2]$ by using derivative find max and min of $g$ then result follows.
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