Let $n \in \mathbb{N}$, $n \geq 3$ and $\delta \in \mathbb{R}$ with $\delta \geq 0$. For every given $n$, I would like to determine the range of $\delta$ such that the following inequality holds:
$$\frac{\sin \frac{\pi}{n} + \sqrt{ \sin^{2} \frac{\pi}{n} + 4n^{2} \delta^{2}}}{2n} < 1 - \sqrt{\delta^{2} + \cos^{2}\frac{\pi}{n}} \qquad (*)$$
Where the r.h.s. of $(*)$, is greater than $0$, i.e. I impose $0< 1 - \sqrt{\delta^{2} + \cos^{2}\frac{\pi}{n}}$ for all $n$ and $\delta$.
Actually, to be more precise, for any given $n \geq 3$ I want to determine the $\mathbf{greatest}$ possible $\delta$ such that $(*)$ holds. Note that an 'approximate' bound for $\delta$ would be ok...
Any help would be appreciated!