### Instruction: Compute $\cos 15^\circ.$.### Response: One way to approach this problem is to use the fact that $\cos (a-b) = \cos a \cos b + \sin a \sin b.$
If we let $a = 45^\circ$ and $b = 30^\circ,$ then we have $\cos 15^\circ = \cos (45^\circ - 30^\circ).$
Now we can use the values of $\cos 45^\circ,$ $\cos 30^\circ,$ $\sin 45^\circ,$ and $\sin 30^\circ$ that we know from the unit circle or from memorization.
We have $\cos 45^\circ = \sin 45^\circ = \frac{\sqrt{2}}{2}$ and $\cos 30^\circ = \frac{\sqrt{3}}{2}$ and $\sin 30^\circ = \frac{1}{2}.$
Plugging these into the formula for $\cos (a-b),$ we get $\cos 15^\circ = \frac{\sqrt{2}}{2} \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \frac{1}{2}.$
Simplifying this expression, we get $\cos 15^\circ = \frac{\sqrt{6} + \sqrt{2}}{4}.$