### Instruction: Compute $\cos 15^\circ.$.### Response: I don't recall the exact value of $\cos 15^\circ$, but I know a trigonometric identity that might help: $\cos (a - b) = \cos a \cos b + \sin a \sin b$.
If I can find two angles $a$ and $b$ whose difference is $15^\circ$ and whose cosine and sine values I know, I can use this identity to find $\cos 15^\circ$.
One possibility is to let $a = 45^\circ$ and $b = 30^\circ$.
I know that $\cos 45^\circ = \sin 45^\circ = \frac{\sqrt{2}}{2}$ and $\cos 30^\circ = \frac{\sqrt{3}}{2}$ and $\sin 30^\circ = \frac{1}{2}$.
Plugging these values into the identity, I get $\cos 15^\circ = \cos (45^\circ - 30^\circ) = \cos 45^\circ \cos 30^\circ + \sin 45^\circ \sin 30^\circ$.
Simplifying, I get $\cos 15^\circ = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$.
Combining the fractions, I get $\cos 15^\circ = \frac{\sqrt{6} + \sqrt{2}}{4}$.