I'm working on a special kind of graphs. To prove some uniqueness, I need to prove that the polynomial \begin{equation} x^{8}-7x^{6}+14x^{4}-8x^{2}+1 \end{equation} does not have any root of the form \begin{equation} 2\cos\frac{(2k+1)\pi}{2n} \quad k\in \lbrace 0,1,\cdots , n-1 \rbrace , n \in \mathbb{E}. \end{equation} Can anybody help me?

Bests.