Simplify
\[\frac{\cos x}{1 - \sin x} - \frac{\cos x}{1 + \sin x}.\]
We can write
\begin{align*}
\frac{\cos x}{1 - \sin x} - \frac{\cos x}{1 + \sin x} &= \frac{\cos x (1 + \sin x)}{(1 - \sin x)(1 + \sin x)} - \frac{\cos x (1 - \sin x)}{(1 + \sin x)(1 - \sin x)} \\
&= \frac{\cos x (1 + \sin x)}{1 - \sin^2 x} - \frac{\cos x (1 - \sin x)}{1 - \sin^2 x} \\
&= \frac{\cos x (1 + \sin x)}{\cos^2 x} - \frac{\cos x (1 - \sin x)}{\cos^2 x} \\
&= \frac{1 + \sin x}{\cos x} - \frac{1 - \sin x}{\cos x} \\
&= \frac{2 \sin x}{\cos x} \\
&= \boxed{2 \tan x}.
\end{align*}