The value of ${\sin }^{-1}\left(\cos \frac{13\pi }{5}\right)\text{ is: }$

${\sin }^{-1}\left(\cos \frac{13\pi }{5}\right)={\sin }^{-1}\left(\cos \left(2\pi +\frac{3\pi }{5}\right)\right)={\sin }^{-1}\left(\cos \frac{3\pi }{5}\right)$
Since, ${\sin }^{-1}x+{\cos }^{-1}x=\frac{\pi }{2}$
$\begin{array}{l}{\sin }^{-1}\left(\cos \frac{3\pi }{5}\right)=\frac{\pi }{2}-{\cos }^{-1}\left(\cos \frac{3\pi }{5}\right)\\ =\frac{\pi }{2}-\frac{3\pi }{5}\\ =-\frac{\pi }{10}\end{array}$