### Instruction: Compute $\cos 72^\circ.$.### Response: I recall that there is a formula for $\cos 36^\circ$ in terms of the golden ratio $\phi = \frac{1 + \sqrt{5}}{2}.$
Specifically, $\cos 36^\circ = \frac{\sqrt{5} + 1}{4}.$
I wonder if I can use this formula to find $\cos 72^\circ.$
I notice that $72^\circ = 2 \cdot 36^\circ,$ so I can use the double angle formula for cosine: $\cos 2\theta = 2 \cos^2 \theta - 1.$
Plugging in $\theta = 36^\circ,$ I get $\cos 72^\circ = 2 \cos^2 36^\circ - 1.$
Now I can substitute $\cos 36^\circ = \frac{\sqrt{5} + 1}{4}$ and simplify.
I get $\cos 72^\circ = 2 \left( \frac{\sqrt{5} + 1}{4} \right)^2 - 1.$
Expanding the square, I get $\cos 72^\circ = 2 \left( \frac{5 + 2\sqrt{5} + 1}{16} \right) - 1.$
Combining the fractions, I get $\cos 72^\circ = \frac{6 + 2\sqrt{5}}{8} - \frac{8}{8}.$
Subtracting, I get $\cos 72^\circ = \frac{-2 + 2\sqrt{5}}{8}.$
I can factor out a 2 from the numerator and simplify.
I get $\cos 72^\circ = \frac{\sqrt{5} - 1}{4}.$