### 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 $\cos 36^\circ = \frac{\sqrt{5} + 1}{4}$.
To verify this formula, I can use the fact that $\cos 72^\circ = \frac{\sqrt{5} - 1}{4}$ and the double angle formula for cosine.
I have $\cos 72^\circ = \cos (2 \cdot 36^\circ) = 2 \cos^2 36^\circ - 1$.
Substituting $\cos 36^\circ = \frac{\sqrt{5} + 1}{4}$, I get $\cos 72^\circ = 2 \cdot \left(\frac{\sqrt{5} + 1}{4}\right)^2 - 1$.
Expanding the square, I get $\cos 72^\circ = \frac{6 + 2 \sqrt{5}}{8} - 1$.
Simplifying, I get $\cos 72^\circ = \frac{\sqrt{5} - 1}{4}$, which matches the other formula.
Therefore, I am confident that $\cos 36^\circ = \frac{\sqrt{5} + 1}{4}$.