### Instruction: Compute $\cos 36^\circ.$.### Response: I recall that there is a formula for $\cos 36^\circ$ in terms of the golden ratio, which is $\frac{1+\sqrt{5}}{2}$.
The formula is $\cos 36^\circ = \frac{\sqrt{5}+1}{4}$.
I could try to prove this formula using geometry, but for now I will just use it as a given fact.