Which of the following is the solution set of the equation $2{\cos }^{-1}x={\cot }^{-1}\left(\frac{2x^2-1}{2x\sqrt{1-x^2}}\right)?$

$2{\cos }^{-1}x={\cot }^{-1}\left(\frac{2x^2-1}{2x\sqrt{1-x^2}}\right)$

Put$x=\cos \theta$ ; L.H.S. = $2\theta$ where   $0\le \theta \le \pi$and $-1\le x\le 1$ ………  (1)

R.H.S. = ${\cot }^{-1}\left(\frac{\cos 2\theta }{2\cos \theta \left|\sin \theta \right|}\right)={\cot }^{-1}\left(\cot 2\theta \right)=2\theta$    if  $0<2\theta <\pi$     ………….  (2)

 From (1) and (2), we get      $0<\theta <\pi /2$                   

  $\therefore x\in (0,1)$