TITLE: Complex numbers equation problem
QUESTION [5 upvotes]: I've been having some trouble with this complex question (not my best topic), and I was wondering if I could get any hints or explainations on how to do it.
Prove that all the roots of the equation $$z^n\cos(n\alpha)+z^{n-1}\cos((n-1)\alpha)+z^{n-2}\cos((n-2)\alpha)+\cdots+z\cos(\alpha)=1,$$ where $\alpha$ is real, lie outside the circle $|z|=\dfrac 12$.
Any help would be greatly appreciated

REPLY [7 votes]: Suppose that there exists a root $z_0$ of the equation such that $|z_0| \le \frac{1}{2}$. Now we have that $$1 = |z_0^n\cos(n\alpha) + z_0^{n-1}\cos((n-1)\alpha) + \dots + z_0\cos(\alpha)|\le$$
$$|z_0^n\cos(n\alpha)| + |z_0^{n-1}\cos((n-1)\alpha)| + \dots + |z_0\cos(\alpha)|\le|z_0^n| + |z_0^{n-1}| + \dots + |z_0|\le$$ $$\frac{1}{2^n} + \frac{1}{2^{n-1}} + \dots + \frac{1}{2} = 1 - \frac{1}{2^n}< 1 $$
Contradiction.