Compute $\tan 15^\circ.$
From the angle subtraction formula,
\begin{align*}
\tan 15^\circ &= \tan (60^\circ - 45^\circ) \\
&= \frac{\tan 60^\circ - \tan 45^\circ}{1 + \tan 60^\circ \tan 45^\circ} \\
&= \frac{\sqrt{3} - 1}{1 + \sqrt{3}} \\
&= \frac{(\sqrt{3} - 1)(\sqrt{3} - 1)}{(\sqrt{3} + 1)(\sqrt{3} - 1)} \\
&= \frac{3 - 2 \sqrt{3} + 1}{2} \\
&= \boxed{2 - \sqrt{3}}.
\end{align*}