We denote by $J_n^{(\mathrm{s})}$ the matrix of $\mathcal{M}_n(\mathbb{R})$ defined by
$$\forall (i,j) \in \llbracket 1,n \rrbracket^2, \quad J_n^{(\mathrm{S})}(i,j) = \frac{2}{\sqrt{2n+1}} \sin\left(\frac{2\pi ij}{2n+1}\right).$$
Show that, for all $p \in \mathbb{N}^*$ and $x \in \mathbb{R} \backslash \pi\mathbb{Z}$,
$$\sum_{k=1}^{p} \cos(2kx) = \frac{1}{2}\left(\frac{\sin((2p+1)x)}{\sin(x)} - 1\right)$$