Question

The value of the expression $sin^{-1}\left(sin \frac{22\pi}{7}\right) cos^{-1}\left(cos \frac{5\pi}{3}\right) + tan^{-1}\left(tan \frac{5\pi}{7}\right) + sin^{-1}\left(cos \,2\right)$ is

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

Answer 1

Answer: (A)

$sin^{-1} sin\left(\frac{22\pi}{7}\right) = sin^{-1} sin\left(3\pi + \frac{\pi}{7}\right) =- \frac{\pi}{7}$
$cos^{-1} cos\left(\frac{5\pi}{3}\right) = cos^{-1} cos\left(2\pi - \frac{\pi}{3}\right) = \frac{\pi}{3} $
$tan^{-1} tan \left(\frac{5\pi}{7}\right) = tan^{-1}tan\left(\pi - \frac{2\pi}{7}\right) = -\frac{2\pi}{7} $
$sin^{-1} cos\left(2\right) = \frac{\pi}{2}-cos^{-1} cos 2 = \frac{\pi}{2} -2$
$ \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 $
$=\frac{17\pi}{42}-2$