QPaperGen

We have:
\begin{align*}
\tan 15^\circ &= \tan(45^\circ - 30^\circ) \\
&= \frac{\tan 45^\circ - \tan 30^\circ}{1 + \tan 45^\circ \cdot \tan 30^\circ} \\
&= \frac{1 - \frac{1}{\sqrt{3}}}{1 + 1 \cdot \frac{1}{\sqrt{3}}} \\
&= \frac{\frac{\sqrt{3} - 1}{\sqrt{3}}}{\frac{\sqrt{3} + 1}{\sqrt{3}}} \\
&= \frac{\sqrt{3} - 1}{\sqrt{3} + 1}
\end{align*}

Now rationalizing the denominator:
\begin{align*}
\tan 15^\circ &= \frac{(\sqrt{3} - 1)^2}{(\sqrt{3} + 1)(\sqrt{3} - 1)} \\
&= \frac{3 + 1 - 2\sqrt{3}}{3 - 1} \\
&= \frac{4 - 2\sqrt{3}}{2} \\
&= 2 - \sqrt{3}
\end{align*}