Solution

Let \(x = sec\theta\), then \(\sqrt{x^2-1} = \sqrt{sec^2\theta-1} = tan\theta\). Therefore, \(cot^{-1}(\frac{1}{\sqrt{x^2-1}}) = cot^{-1}(cot\theta) = \theta = sec^{-1}x\), which is the simplest form.