Compute $\cos 15^\circ.$
From the angle subtraction formula,
\begin{align*}
\cos 15^\circ &= \cos (60^\circ - 45^\circ) \\
&= \cos 60^\circ \cos 45^\circ + \sin 60^\circ \sin 45^\circ \\
&= \frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} \\
&= \boxed{\frac{\sqrt{2} + \sqrt{6}}{4}}.
\end{align*}