Range of  ${\sin }^{-1}x-{\cos }^{-1}x$  is

$$\begin{gathered} Si{n}^{-1}x-Co{s}^{-1}x=\frac{\pi }{2}-{\cos }^{-1}x-{\cos }^{-1}x \\ =\frac{\pi }{2}-2{\cos }^{-1}x W.K.T. \Rightarrow 0\le {\text{Cos}}^{-1}x\le \pi \\ 0\le 2{\text{Cos}}^{-1}x\le 2\pi \Rightarrow -2\pi \le -2{\text{Cos}}^{-1}x\le 0 \\ \Rightarrow \frac{\pi }{2}-2\pi \le \frac{\pi }{2}-2{\text{Cos}}^{-1}x\le \frac{\pi }{2} \Rightarrow -\frac{3\pi }{2}\le \frac{\pi }{2}-2{\text{Cos}}^{-1}x\le \frac{\pi }{2} \end{gathered}$$