Question

The value of the expression $si{n}^{-1}\left(sin\frac{22\pi }{7}\right)co{s}^{-1}\left(cos\frac{5\pi }{3}\right)+ta{n}^{-1}\left(tan\frac{5\pi }{7}\right)+si{n}^{-1}\left(cos2\right)$ is

• A. $\frac{17\pi }{42}-2$
• B. $-2$
• C. $\frac{-\pi }{21}-2$
• D. none of these

Answer 1

Answer: (A)

$si{n}^{-1}sin\left(\frac{22\pi }{7}\right)=si{n}^{-1}sin\left(3\pi +\frac{\pi }{7}\right)=-\frac{\pi }{7}$
$co{s}^{-1}cos\left(\frac{5\pi }{3}\right)=co{s}^{-1}cos\left(2\pi -\frac{\pi }{3}\right)=\frac{\pi }{3}$
$ta{n}^{-1}tan\left(\frac{5\pi }{7}\right)=ta{n}^{-1}tan\left(\pi -\frac{2\pi }{7}\right)=-\frac{2\pi }{7}$
$si{n}^{-1}cos\left(2\right)=\frac{\pi }{2}-co{s}^{-1}cos2=\frac{\pi }{2}-2$
$\therefore$ Required value $=-\frac{\pi }{7}+\frac{\pi }{3}-\frac{2\pi }{7}+\frac{\pi }{2}-2$
$=\frac{\left(-18+35\right)\pi }{42}-2$
$=\frac{17\pi }{42}-2$