Write the function in the simplest form:

$\tan ^{-1} \frac{1}{\sqrt{x^{2}-1}},|x|>1$

Put $x=\operatorname{cosec} \theta \Rightarrow \theta=\operatorname{cosec}^{-1} x$

$\therefore \tan ^{-1} \frac{1}{\sqrt{x^{2}-1}}=\tan ^{-1} \frac{1}{\sqrt{\operatorname{cosec}^{2} \theta-1}}$

$=\tan ^{-1}\left(\frac{1}{\cot \theta}\right)=\tan ^{-1}(\tan \theta)$

$=\theta=\operatorname{cosec}^{-1} x=\frac{\pi}{2}-\sec ^{-1} x \quad\left[\operatorname{cosec}^{-1} x+\sec ^{-1} x=\frac{\pi}{2}\right]$