Trying to resolve a trig identity

[Dustinsfl]

I am trying to resolve a trig identity for some notes I am typing up. On paper, I wrote recall $e(\sin(E_1) - \sin(E_0)) = 2\cos(\zeta)\sin(E_m)$. I have no idea what I am recalling this from now at least.

Identities I have set up are:

\begin{align}
E_p &= \frac{1}{2}(E_1 + E_2)\\
E_m &= \frac{1}{2}(E_1 - E_2)\\
x &= a\cos(E)\\
y &= a\sqrt{1 - e^2}\sin(E)\\
\cos(\zeta) &= e\cos(E_p)\\
\alpha &= \zeta + E_m\\
\beta &= \zeta - E_m
\end{align}

Lambert Section this may be easier to understand if you look at it.

 

[MarkFL]

Let's begin with the left side (but write it instead as):

\(\displaystyle e\left(\sin\left(E_1 \right)-\sin\left(E_2 \right) \right)\)

Using a sum-to-product identity, we may write this as:

\(\displaystyle 2e\sin\left(\frac{E_1-E_2}{2} \right)\cos\left(\frac{E_1+E_2}{2} \right)\)

Now using the identities you have set up, this becomes:

\(\displaystyle 2\cos(\zeta)\sin\left(E_m \right)\)