Question

The principal value of ${\cos }^{-1}\left\{\frac{1}{\sqrt{2}}(\cos \frac{9\pi }{10}-\sin \frac{9\pi }{10})\right\}$ is

• A. $-\frac{3\pi }{20}$
• B. $-\frac{7\pi }{20}$
• C. $-\frac{7\pi }{10}$
• D. none of these

Answer 1

The correct option is D none of these

$co{s}^{-1}\left[\frac{1}{\sqrt{2}}cos\right(\frac{9\pi }{10})-\frac{1}{\sqrt{2}}sin(\frac{9\pi }{10}\left)\right]$

$=co{s}^{-1}\left[cos{45}^{0}cos\right(\frac{9\pi }{10})-sin{45}^{0}sin(\frac{9\pi }{10}\left)\right]$

$=co{s}^{-1}\left[cos\right(\frac{\pi }{4}+\frac{9\pi }{10}\left)\right]$

$=co{s}^{-1}\left[cos\right(\frac{23\pi }{20}\left)\right]$

$=co{s}^{-1}\left[cos\right(\pi +\frac{3\pi }{20}\left)\right]$

$=co{s}^{-1}[-cos(\frac{3\pi }{20}\left)\right]$

$=\pi -co{s}^{-1}\left[cos\right(\frac{3\pi }{20}\left)\right]$

$=\pi -\frac{3\pi }{20}$

$=\frac{17\pi }{20}$