Simplify : $\displaystyle \cos ^{-1} x+\cos ^{-1}\left[\frac{x}{2}+\frac{\sqrt{3-3 x^{2}}}{2}\right] ; \frac{1}{2} \leq x \leq 1$

Let $\displaystyle \mathbf{x} \boldsymbol{=} \boldsymbol{\operatorname { c o s }} \boldsymbol{\theta}$,\begin{aligned} & \cos ^{-1} x+\cos ^{-1}\left[\frac{x}{2}+\frac{\sqrt{3-3 x^{2}}}{2}\right] = & \theta+\cos ^{-1}\left[\frac{\cos \theta}{2}+\frac{\sqrt{3}}{2} \times \sin \theta\right] = & \theta+\cos ^{-1}\left[\cos \left(\frac{\pi}{3}-\theta\right)\right]=\theta+\frac{\pi}{3}-\theta=\frac{\pi}{3} \end{aligned} $$