Question

${\cos }^{-1}\left\{\frac{1}{\sqrt{2}}(\cos \frac{7\pi }{5}-\sin \frac{2\pi }{5})\right\}$ is equal to

• A. $\frac{23\pi }{20}$
• B. $\frac{13\pi }{20}$
• C. $\frac{33\pi }{20}$
• D. $\frac{17\pi }{20}$

Answer 1

The correct option is D $\frac{17\pi }{20}$

${\cos }^{-1}\left(\frac{1}{\sqrt{2}}\right(\cos (\pi +\frac{2\pi }{5})-\sin \left(\frac{2\pi }{5}\right)\left)\right)$
$={\cos }^{-1}\left(\frac{1}{\sqrt{2}}\right(-\cos \left(\frac{2\pi }{5}\right)-\sin \left(\frac{2\pi }{5}\right)\left)\right)$
$=\pi -{\cos }^{-1}\left(\frac{1}{\sqrt{2}}\right(\cos \left(\frac{2\pi }{5}\right)+\sin \left(\frac{2\pi }{5}\right)\left)\right)$
$=\pi -{\cos }^{-1}\left(\cos \right(\frac{2\pi }{5}-\frac{\pi }{4}\left)\right)$
$=\pi -{\cos }^{-1}\left(\cos \right(\frac{3\pi }{20}\left)\right)$
$=\pi -\frac{3\pi }{20}$
$=\frac{17\pi }{20}$