Determine $z^n+z^{-n}$ if $z+\frac{1}{z}=-2\cos{x}$ Determine $z^n+z^{-n}$ if $z+\frac{1}{z}=-2\cos{x}$ with $z \in \mathbb{C}$.

Since $z + \frac{1}{z} = - 2 \cos(x)$ is equivalent to $z^2 + 2 z \cos(x) + 1 = 0$, it is solved by $z_{1,2} = -\cos(x) \pm i \sqrt{1-\cos^2(x)}$. Since $1-\cos^2(x) = \sin^2(x)$, these also solve the equation $\tilde{z}_{1,2} = -\cos(x) \mp i \sin(x) = -\exp(\pm i x)$.
Now to find $z^n+z^{-n}$ for $z$ being the solution of $z+\frac{1}{z} = -2 \cos(x)$ subsitute the $z = \tilde{z}_{1,2}$.
