Constructible $2^{2n + 2}$-th roots of unity

Question

This is a problem from Dummit and Foote: Section 14.5 Exercise 8(c). Let $\zeta_{2^{n+2}}$ be a primitive $2^{n+2}$-th root of unity, and similarly let $\zeta_{2^{n+3}}$ be a primitive $2^{n+3}$-th root of unity. If $\alpha_n = \zeta_{2^{n+2}} + \zeta_{2^{n+2}}^{-1}$ and $\alpha_{n+1} = \zeta_{2^{n+3}} + \zeta_{2^{n+3}}^{-1}$ then show that $\alpha_{n+1}^2 = 2 + \alpha_n$ and conclude that $$\alpha_n = \sqrt{2 + \sqrt{2 + \sqrt{ \cdots + \sqrt{2}}}}$$ ($n$ times). I see that $\alpha_{n+1}^2 = 2 +\alpha_n$, but I have no idea how to show the conclusion regarding the explicit form of $\alpha_n$. Any suggestions would be much appreciated!