Proof: $\sin\left(\frac{x-x_0}{2}\right)\sin\left(\frac{x-x_1}{2}\right) = \frac{1}{2}\cos \frac{x_1-x_0}{2} - \frac{1}{2}\cos(x-\frac{x_1+x_0}{2})$

Question

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!

Answer 1

$\displaystyle\cos(A-B)-\cos(A+B)=\cdots=2\sin A\sin B$ (Werner Formula)

Can you recognize $A,B$ here?

---

Alternatively,

We can use Prosthaphaeresis Formula $\displaystyle\cos C-\cos D=2\sin\frac{C+D}2\sin\frac{D-C}2$ as well.

Observe that Prosthaphaeresis formula can be derived from Werner's and vice versa