### Instruction: Compute $\cos 72^\circ.$.### Response: I don't know the exact value of $\cos 72^\circ,$ but I do know some other angles whose cosine values are easy to remember, such as $0^\circ, 30^\circ, 45^\circ, 60^\circ,$ and $90^\circ.$
Maybe I can use some trigonometric identity or formula to relate $\cos 72^\circ$ to some of these angles.
One identity that comes to mind is the double-angle formula for cosine: $\cos 2\theta = 2\cos^2 \theta - 1.$
If I apply this formula with $\theta = 36^\circ,$ I get $\cos 72^\circ = 2\cos^2 36^\circ - 1.$
Now I have reduced the problem to finding $\cos 36^\circ.$
Unfortunately, I don't know the value of $\cos 36^\circ$ either.
Is there another way to find it?
I recall another identity that involves cosines of angles in a regular pentagon.
It says that $\cos 36^\circ = \frac{\sqrt{5} + 1}{4}.$
This identity can be derived by drawing a regular pentagon and using the law of cosines on one of its triangles, but I won't go into the details here.
I'll just trust that it is true.
Now I can substitute this value into the double-angle formula and simplify: $\cos 72^\circ = 2\left(\frac{\sqrt{5} + 1}{4}\right)^2 - 1 = \frac{\sqrt{5} + 1}{4} - \frac{1}{2} = \frac{\sqrt{5} - 1}{4}.$
This is the exact value of $\cos 72^\circ.$