I looked up a proof for trigonometric interpolation, and it works with this equation:
\begin{align} \sin\left(\frac{x-x_0}{2}\right)\sin\left(\frac{x-x_1}{2}\right) = \frac{1}{2}\cos \frac{x_1-x_0}{2} - \frac{1}{2}\cos\left(x-\frac{x_1+x_0}{2}\right) \end{align}
where $x_0, x_1$ are some known real $x$-values. Does anyone know what trigonometric identities could be used to see that the equation is correct? I really don't know where to begin. Thank you!