Let ABC be an isosceles triangle with A=72 degrees, B=36 degrees, C=72 degrees. Let AB = BC = 1 and CA = x. We will solve for x with a simple geometric argument.

Draw the angle bisector from A intersecting BC at P. Observe that angle CAP is 36 degrees, so angle APC is 72 degrees, so triangle CAP is another 72-36-72 isosceles triangle. Therefore AP = x.

Observe that angle PAB is 36 degrees, so triangle BPA is isosceles, so BP also = x and PC = 1-x.

Since triangle ABC is similar to triangle CAP (both being 36-72-36 isosceles), we have PC/CA = CA/AB, or (1-x)/x = x/1. This is the formula for the golden ratio; “the smaller is to the larger as the larger is to the whole”… We wind up with x = 1/φ.

Then just apply the Law of Cosines to triangle ABC to get the cosine of 36 degrees in terms of φ. It’s φ/2 if I recall correctly.