Spoiler: hints ahead.....
The key step is finding a simple expression for sin 18. This is the essence of the problem. Once you derive an expression for sin 18, the rest is some obvious algebra.
Accomplishing the first step directly (even from first principles) seemed for me improbable. But, I found a derivation and I give the barest outline of it here. This may help someone else to solve it:
Start with noticing 5 x 18 = 90, and then use trig
double and triple angle formulae (since 5 = 2 + 3). Next, a difference trig identity leads one to finally solve a quadratic, to gives one
sin 18.
This method for getting sin 18 may be found in full detail
here.
Edited on September 18, 2023, 3:08 am
proof found
