Answer
$\frac{1}{\sqrt{x^2-1}}$
Work Step by Step
Assume that $\sec\theta =\frac{x}{1}$, then by solving the triangle, we have:
$$\cot \theta=\frac{1}{\sqrt{x^2-1}}.$$
Now, since $\theta =\sec^{-1}x $, we have
$$\cot(\sec^{-1}x)=\cot\theta=\frac{1}{\sqrt{x^2-1}}.$$
