How to show that 1/sin(xi-yj) obey Cauchy determinants Identity

Question

This question arose while I was following an article on the direction of bosonization: Functional Integral Bosonization for Impurity in Luttinger Liquid. There is an identity in the article that caused my doubts: $$ \begin{aligned} \operatorname{det} \frac{1}{\sin \left(z_i-z_j^{\prime}\right)}=(-1)^{n(n-1) / 2} \frac{\prod_{i<j}^n \sin \left(z_i-z_j\right) \sin \left(z_i^{\prime}-z_j^{\prime}\right)}{\prod_{i, j=1}^n \sin \left(z_i-z_j^{\prime}\right)}. \end{aligned} $$ I know the standard Cauchy Determinant should be： $$ \operatorname{det}\left(C_n\right)=\left[\begin{array}{cccc} \frac{1}{x_1-y_1} & \frac{1}{x_1-y_2} & \cdots & \frac{1}{x_1-y_n} \\ \frac{1}{x_2-y_1} & \frac{1}{x_2-y_2} & \cdots & \frac{1}{x_2-y_n} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{1}{x_n-y_1} & \frac{1}{x_n-y_2} & \cdots & \frac{1}{x_n-y_n} \end{array}\right], $$ and the result is $$ \operatorname{det}\left(C_n\right)= (-1)^{n(n-1)/2} \frac{\prod_{1 \leq i<j \leq n}\left(x_i-x_j\right)\left(y_i-y_j\right)}{\prod_{1 \leq i, j \leq n}\left(x_i-y_j\right)}. $$ So my first reaction is whether the formula in the text is wrong, after all, the form of the function is completely different from the standard equation. So I did a simple verification with Mathematica, and surprisingly, the formula in the text is completely correct when n=2,3,4! I have no doubt that there is some kind of mathematical induction that can prove that the result is true when n is any positive integer. But what I want to ask more is whether there is systematic research on this kind of promotion? What properties should a generalized function have to allow a similar generalization to hold? If there is, I hope to point it out, thank you very much.