Let $[\theta_1,\theta_2, \dots, \theta_N]^\mathrm{T} \, \in \mathbb{R}^N$. The angles are not all identical (on the circle), i.e. $[\theta_1,\theta_2, \dots, \theta_N] \not \equiv c [1,1,\dots, 1]^\mathrm{T}\,\, \mathrm{mod}\,\, 2\pi$. Define matrices $C$ and $S$ as:

\begin{align} \begin{split} [C]_{jl}&= 1 \,\, \mathrm{if}\,\, j =l \\ [C]_{jl}&= \cos(\theta_j-\theta_l)\,\, \mathrm{if} \,\, j \neq l \end{split} \end{align}

and

\begin{align} \begin{split} [S]_{jl}&= 0 \,\, \mathrm{if}\,\, j =l \\ [S]_{jl}&= \sin(\theta_j-\theta_l)\,\,\mathrm{if} \,\, j \neq l\,. \end{split} \end{align}

Is it true that any $v$ that belongs to the nullspace of S also belongs to the nullspace of C?

My repeated simulations in MATLAB with randomly generated $[\theta_1,\theta_2, \dots, \theta_N]^\mathrm{T}$ seems to suggest that $C$ and $S$ are rank $2$ and any eigenvectors with $0$ eigenvalue for one is an eigenvector with $0$ eigenvalue for the other.