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

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!

• all trig identities can be proven using Euler's formula. Are you allowed to use it? – Alex May 24 '14 at 13:22

$\displaystyle\cos(A-B)-\cos(A+B)=\cdots=2\sin A\sin B$ (Werner Formula)
Can you recognize $A,B$ here?
We can use Prosthaphaeresis Formula $\displaystyle\cos C-\cos D=2\sin\frac{C+D}2\sin\frac{D-C}2$ as well.