Evaluate cos[cos-1(-√3/2) + π/6]

Question:

Evaluate cos[cos-1(-√3/2) + π/6]

Solution:

$\cos \left[\cos ^{-1}\left(\frac{-\sqrt{3}}{2}\right)+\frac{\pi}{6}\right]$

$=\cos \left[\cos ^{-1}\left(\cos \frac{5 \pi}{6}\right)+\frac{\pi}{6}\right]$ $\left(\because \cos \frac{5 \pi}{6}=\frac{-\sqrt{3}}{2}\right)$

$=\cos \left(\frac{5 \pi}{6}+\frac{\pi}{6}\right)$ $\left(\because \cos ^{-1} \cos x=x ; x \in[0, \pi]\right)$

$=\cos (\pi)=-1$

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