Vector geometry proofs

Question

Prove that all the roots of the equation $$z^n \cos(n \alpha)+z^{n-1} \cos((n-1) \alpha)+ \cdots +z \cos(\alpha)=1$$, where alpha is real, lie outside the circle $|z|=1/2$. How do I approach this question?

Answer 1

Hint.

Using de Moivre's identity and calling $Z = z(\cos\alpha + i\sin\alpha)$ we arrive at

$$ \frac{Z^{n+1}-1}{Z-1}-1=1\Rightarrow \mbox{Re}(2Z-Z^{n+1})=1 $$

or equivalently

$$ 2|Z|\cos\alpha = 1+|Z|^{n+1}\cos((n+1)\alpha) $$

Answer 2

let $z=re^{i\beta}$ then $$ \cos(m\alpha)=\frac{e^{im\alpha}+e^{-im\alpha}}{2} $$ and you calculate the sum $$ 1=\sum_{m=1}^{n}{z\cos(m\alpha)} $$ (this sum is sum two geometric progresion sum) and $Re(1)=1,\>Im(1)=0$.