Question

The value of $\cos \left[\frac{1}{2}{\cos }^{-1}\cos \left(-\frac{14\pi }{5}\right)\right]$ is

• A. $\cos \left(-\frac{7\pi }{5}\right)$
• B. $\sin \left(-\frac{\pi }{10}\right)$
• C. $\cos \left(\frac{2\pi }{5}\right)$
• D. $-\cos \left(\frac{3\pi }{5}\right)$

Answer 1

The correct option is C $\cos \left(\frac{2\pi }{5}\right)$

Given

${\cos }^{-1}\cos \left(\frac{-14\pi }{5}\right)$

${\cos }^{-1}\cos \left(\frac{14\pi }{5}\right)$

${\cos }^{-1}\cos (2\pi +\frac{4\pi }{5})$

${\cos }^{-1}\cos \left(\frac{4\pi }{5}\right)$

$(\cos (2\pi +x)=\cos x)$

$⟹\frac{4\pi }{5}$

$\cos \left[\frac{1}{2}{\cos }^{-1}\cos \left(\frac{-14\pi }{5}\right)\right]$

$⟹\cos [\frac{1}{2}\times \frac{4\pi }{5}]$

$⟹\cos \left(\frac{2\pi }{5}\right)$