The value of  ${\sin }^{-1}\left[\cos \{{\cos }^{-1}\left(\cos x\right)+{\sin }^{-1}\left(\sin x\right)\}\right]$where  $x\in (\frac{\pi }{2},\pi )$is equal to 

${\sin }^{-1}\left[\cos \{{\cos }^{-1}\left(\cos x\right)+{\sin }^{-1}\left(\sin x\right)\}\right]$

$={\sin }^{-1}\left[\cos (x+\pi -x)\right]$ $$\begin{gathered} \left(\because {\sin }^{-1}\left(\mathrm{sinx}\right)={\sin }^{-1}\left(\sin \left(\mathrm{\pi }-\mathrm{x}\right)\right)=\mathrm{\pi }-\mathrm{x} \\ \left(\because \mathrm{x}\in \left(\frac{\mathrm{\pi }}{2},\mathrm{\pi }\right)\Rightarrow \frac{\mathrm{\pi }}{2}<\mathrm{x}<\mathrm{\pi }\Rightarrow 0<\mathrm{\pi }-\mathrm{x}<\frac{\mathrm{\pi }}{2}\right)\right) \end{gathered}$$

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