Compute $\cos 75^\circ.$
From the angle addition formula,
\begin{align*}
\cos 75^\circ &= \cos (45^\circ + 30^\circ) \\
&= \cos 45^\circ \cos 30^\circ - \sin 45^\circ \sin 30^\circ \\
&= \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2} \\
&= \boxed{\frac{\sqrt{6} - \sqrt{2}}{4}}.
\end{align*}