I need to solve the below mentioned equation and try to find a unique solution for $\epsilon$ for the range between (-1,1) in terms of $n$.
$\begin{equation} \sum_{j=1}^{n-1} \frac{(2{\alpha_j} + 1)( \alpha_j -1)}{{\bigg[(\alpha_j - 1) \bigg((2{\alpha_j}+3) - \epsilon(2{\alpha_j}+1)\bigg) \bigg]\,\,\,}^{2}} = 0 \end{equation} $
where
$\alpha_j=\cos\frac{2{\pi}j}{n}$.
With simulations, I can see that there's a unique solution for $\epsilon$ when $\epsilon$ is in (-1,1) and I need to find that as an expression in terms of $n$. Any suggestions