The value of $\lim\limits_{x \rightarrow \frac{\pi}{2}} \frac{\sin ^{-1}(1 - cosec ~x)}{\frac{\pi}{2}-x}$ is usual to

0

$\lim\limits_{x \rightarrow \frac{\pi}{2}} \frac{\sin ^{-1}(1 - cosec ~x)}{\frac{\pi}{2}-x}=\frac{0}{0}form$

Applying L'Hopital's Rule

$\lim\limits_{x \rightarrow \frac{\pi}{2}}\frac{cosec\,x\cot x}{-\sqrt{1-(1-cosec)^2}}=0$