Question

Which of the following is/are the value of $\cos [\frac{1}{2}co{s}^{-1}(cos(\frac{-14\pi }{5}))]$

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

Answer 1

Answer: (B), (C), (D)

The correct options are
B $\sin \frac{\pi }{10}$
C $\cos \frac{2\pi }{5}$
D $-\cos \frac{3\pi }{5}$

$\cos (\frac{-14\pi }{5})=\cos (\frac{14\pi }{5})=\cos (2\pi +\frac{4\pi }{5})=\cos (\frac{4\pi }{5})$
now,

$\cos [\frac{1}{2}co{s}^{-1}(cos(\frac{-14\pi }{5}))]=\cos [\frac{1}{2}co{s}^{-1}(cos(\frac{4\pi }{5}))]=\cos \left[\frac{1}{2}\times \frac{4\pi }{5}\right]=\cos \frac{2\pi }{5}$
also,
$\cos \frac{2\pi }{5}=\sin (\frac{\pi }{2}-\frac{2\pi }{5})=\sin \frac{\pi }{10}$
also,
$\cos \frac{2\pi }{5}=-\cos (\pi -\frac{2\pi }{5})=-\cos \frac{3\pi }{5}$