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