### Instruction: Compute $\cos 72^\circ.$.### Response: So we know that $\cos 72^\circ = \cos (2 \cdot 36^\circ)$.
And since we know that $\cos 2 \theta = 2 \cos^2 \theta - 1$, we can substitute to get $\cos 72^\circ = 2 \cos^2 36^\circ - 1$.
But we also know that $\cos 36^\circ = \frac{\sqrt{5} + 1}{4}$.
So we can substitute again to get $\cos 72^\circ = 2 \cdot \left( \frac{\sqrt{5} + 1}{4} \right)^2 - 1$.