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If $\frac{1}{2} \leq x \leq 1$, then $\cos ^{-1} x+\cos ^{-1}\left(\frac{x}{2}+\frac{1}{2} \sqrt{3-3 x^2}\right)$ is equal to
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$\frac{\pi}{3}$
$\begin{aligned} & \therefore \cos ^{-1} x+\cos ^{-1}\left(\frac{x}{2}+\frac{\sqrt{3}}{2} \sqrt{1-x^2}\right) \\ & =\cos ^{-1} x+\cos ^{-1}\left(x \times \frac{1}{2}+\frac{\sqrt{3}}{2} \times \sqrt{1-x^2}\right) \\ & =\cos ^{-1} x+\cos ^{-1}\left(\frac{1}{2}\right)-\cos ^{-1} x \\ & =\cos ^{-1}\left(\frac{1}{2}\right)=\frac{\pi}{3}\end{aligned}$
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