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

$$\begin{gathered} {\sin }^{-1}\left(\mathrm{x}\right)\text{ is defined in }\mathrm{x}\in [-1,1] \\ {\cos }^{-1}\left(x\right)\text{ is defined in }x\in [-1,1] \\ {\tan }^{-1}\left(\mathrm{x}\right)\text{ is defined in }\mathrm{x}\in (-\mathrm{\infty },+\mathrm{\infty }) \\ \text{ So }f\left(x\right)={\sin }^{-1}\left(x\right)+{\cos }^{-1}\left(x\right)+{\tan }^{-1}\left(x\right)\text{ is defined in }x\in [-1,1] \\ \mathrm{So}\mathrm{f}\left(\mathrm{x}\right)=\frac{\pi }{2}+{\tan }^{-1}\left(\mathrm{x}\right) \\ \mathrm{Now}{f}^{\mathrm{\prime }}\left(x\right)=\frac{1}{1+x^2}>0\mathrm{\forall }x\in [-1,1] \\ \text{ So }\mathrm{f}(-1)=\frac{\pi }{2}+{\tan }^{-1}(-1)=\frac{\pi }{2}-\frac{\pi }{4}=\frac{\pi }{4} \\ \text{ and }\mathrm{f}(+1)=\frac{\pi }{2}+{\tan }^{-1}\left(1\right)=\frac{\pi }{2}+\frac{\pi }{4}=\frac{3\pi }{4} \\ \mathrm{So}f\left(x\right)\in \left[\frac{\pi }{4},\frac{3\pi }{4}\right] \end{gathered}$$