求证三角函数恒等式

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\[
\frac{\prod_{j=1}^n \sin \left(z-\alpha_j\right)}{\prod_{j=0}^n \sin \left(z-\beta_{j}\right)}=\sum_{k=0}^n \frac{\prod_{j=1}^n \sin \left(\alpha_j-\beta_k\right)}{\sin \left(z-\beta_k\right) \prod_{j \neq k} \sin \left(\beta_{j}-\beta_k\right)}
\]