I have already shown a $10$-gon is constructible. The I am trying to use the fact that the angle $\cos(2\pi/10)$ is constructible and that$$\cos(2\pi/10) = \cos(3(2\pi/30)) = 4\cos^3(2\pi/30)-3\cos(2\pi/30)$$and that $\cos(2\pi/10) = (\sqrt5-1)/8$.
After this step I get lost. All help appreciated.
All you need is to show that a $15$-gon is constructible (you construct it and then you bissect each angle).
In order to prove that you can construct a $15$-gon, you use the fact that you can construct a regular triangle and a regular pentagon. That and the fact that $\dfrac25-\dfrac13=\dfrac1{15}$ is enough to do it.