Question

What is the value of $\cos [{\cos }^{-1}\left(\frac{-\sqrt{3}}{2}\right)+\frac{\pi }{6}]$ ?

Answer 1

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

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

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

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