What is an efficient way of showing that the matrix $$\begin{align} P\triangleq \begin{bmatrix}\cos\theta_1&\sin\theta_1&...&\cos\theta_n&\sin\theta_n\\ \cos2\theta_1&\sin2\theta_1&...&\cos2\theta_n&\sin2\theta_n\\ \vdots&\vdots&~&\vdots&\vdots\\ \cos2n\theta_1&\sin 2n\theta_1&...&\cos2n\theta_n&\sin2n\theta_n\end{bmatrix}\in\mathbb{R}^{2n\times 2n} \end{align}$$ is nonsingular for distinct $\theta_i\in(0,\pi)$ (or similar conditions on $\theta_i$).
I have seen this similar post but I cannot do the same here. Any help is appreciated. Thanks
Let $D=\pmatrix{1&1\\ i&-i}$. Then $$ P(D\oplus D\oplus\cdots\oplus D) =\pmatrix{ e^{i\theta_1}&e^{-i\theta_1}&\cdots&e^{i\theta_n}&e^{-i\theta_n}\\ e^{2i\theta_1}&e^{-2i\theta_1}&\cdots&e^{2i\theta_n}&e^{-2i\theta_n}\\ \vdots&\vdots&&\vdots&\vdots\\ e^{(2n-1)i\theta_1}&e^{-(2n-1)i\theta_1}&\cdots&e^{(2n-1)i\theta_n}&e^{-(2n-1)i\theta_n}\\ e^{2ni\theta_1}&e^{-2ni\theta_1}&\cdots&e^{2ni\theta_n}&e^{-2ni\theta_n}}, $$ and in turn $P(D\oplus D\oplus\cdots\oplus D)\operatorname{diag}(e^{-i\theta_1},e^{i\theta_1},\cdots,e^{-i\theta_n},e^{i\theta_n})$ is the Vandermonde matrix for $e^{i\theta_1},\ e^{-i\theta_1},\ \ldots,\ e^{i\theta_n},\ e^{-i\theta_n}$.