The value of ${\cos ^{ - 1}}\left( {\cos \frac{{3\pi }}{2}} \right)$ is

We have, ${\cos ^{ - 1}}\left( {\cos \frac{{3\pi }}{2}} \right)$

$= {\cos ^{ - 1}}\cos \left( {2\pi - \frac{\pi }{2}} \right)$

$= {\cos ^{ - 1}}\cos \left( {\frac{\pi }{2}} \right) = \frac{\pi }{2}$

Note
Remember that, ${\cos ^{ - 1}}\left( {\cos \frac{{3\pi }}{2}} \right) \ne \frac{{3\pi }}{2}$
$\frac{{3\pi }}{2} \notin (0,\pi )$