I would like to apply the so-called cotangent DVR method (https://doi.org/10.1016/j.chemphys.2010.07.006) for numerical integration, which uses the following basis functions:
$\chi_n(\theta) = \begin{cases} \cos(n\,\theta), & {\rm if}~n\le N - 2, \\ \sin(\theta), & {\rm if}~n= N - 1, \\ \sin(2\theta), & {\rm if}~n= N. \end{cases}$
Using these functions, an overlap matrix $\mathbf{S}=\{s_{jk}\}$ and a coordinate matrix $\mathbf{Q}=\{q_{jk}\}$ can be constructed, where
$s_{jk} = \int\limits_0^\pi \chi_j(\theta) \chi_k(\theta) \sin \theta\,d \theta$
and
$q_{jk} = \int\limits_0^\pi \cot(\theta)\,\chi_j(\theta) \chi_k(\theta) \sin \theta\,d \theta$
The $g_1,g_2,\ldots,g_N$ grid points can be derived by solving the generalized problem $\mathbf{Q}\,\mathbf{T} = \mathbf{S}\,\mathbf{T}\,{\rm diag}(g_1,g_2,\ldots,g_N)$.
For practical reasons, I need to use the transformed $g'_1,g'_2,\ldots,g'_N$ grid points with $g'_i = \cos({\rm acot}(g_i))$ in my calculations.
Assuming that the weight function is $W(\theta)=1$, what would be the quadrature weights of the original and the transformed grid set, $w_1,w_2,\ldots,w_N$ and $w'_1,w'_2,\ldots,w'_N$, respectively? I tried to use the so-called Golub–Welch scheme which yields $w_i = \pi\,t_{1,i}^2$, where $t_{1,i}$ is an entry of the $\mathbf{T}$ matrix, but these weights did not provide the correct integrals.