I have one question which is related to showing that the $R_n$ matrix is degenerate when $n\geq 3$.
$R_n = \begin{pmatrix} 1 & \cos(\lambda) & \cos(2\lambda) & \cos(3\lambda) &... & \cos((n-1)\lambda) \\ cos(\lambda) & 1 & \cos(\lambda) & \cos(2\lambda) & ... & \cos((n-2)\lambda) \\ ... & ... & ... & ... & ... & ... \\ \cos((n-1)\lambda) & \cos((n-2)\lambda) & \cos((n-3)\lambda) & \cos((n-1)\lambda) & ... & 1 \end{pmatrix}$
One way is to show that this matrix is a product of matrix $A$ and its inverse, i.e.
$A = \begin{pmatrix} 1 & \cos(\lambda) & 3\cos(2\lambda) & ... & \cos((n-1)\lambda)\\ 0 & \sin(\lambda) & \sin(2\lambda) & ... & \sin((n-1)\lambda) \end{pmatrix}$
Then we can see that $rang(R_n) \leq min(rang(A^{-1}),rang(A))$. Here $0\leq\lambda\leq \pi.$
Maybe there is another approach to show that the matrix $R_n$ is degenerate when $n\geq3$ ?