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