If $z^n + z^{-n} = 2\cos(n\theta)$, prove $5z^4 - z^3 - 6z^2 - z + 5 = 0$

Question

If $z^n + z^{-n} = 2\cos(n\theta)$, show that $$5z^4 - z^3 - 6z^2 - z + 5 = 0 \Longrightarrow 10\cos^2\theta-\cos \theta -8=0.$$

I've substituting 2, 3 and 4 for $n$ and then playing with the answers i got but i didn't get anywhere.