Question

For the principal value, evaluate the following:

${\displaystyle {\sin }^{-1}\left[\cos \left\{2{\text{cosec}}^{-1}(-2)\right\}\right]}$

Answer 1

${\displaystyle {\sin }^{-1}\left[\cos \left\{2{\text{cosec}}^{-1}(-2)\right\}\right]={\sin }^{-1}\left[\cos \left\{2{\text{cosec}}^{-1}\text{cosec-}\frac{\pi }{6}\right\}\right]}$

${\displaystyle ={\sin }^{-1}\left[\cos \{-\frac{\pi }{3}\}\right]}$

${\displaystyle ={\sin }^{-1}\left[\cos \left(\frac{\pi }{3}\right)\right]}$

${\displaystyle ={\sin }^{-1}\left(\frac{1}{2}\right)}$

${\displaystyle ={\sin }^{-1}\left(\frac{\sin \pi }{6}\right)}$

${\displaystyle =\frac{\pi }{6}}$