How to prove the identity $\sin2x + \sin2y = 2\sin(x + y)\cos(x - y)$ I've been trying to prove the identity $$\sin2x + \sin2y = 2\sin(x + y)\cos(x - y).$$
So far I've used the identities based off of the compound angle formulas. I'm not quite sure if those identities would work with proving the above identity.
Thank you in advance.

\begin{align}
   2\sin(x + y)\cos(x - y)
   &= 2(\sin x \cos y + \cos x \sin y)\cdot
       (\cos x \cos y + \sin x \sin y) \\
   &= 2\sin x \cos x(\cos^2 y + \sin^2 y) + 
      2\sin y \cos y(\cos^2 x + \sin^2 x) \\
   &= 2\sin x \cos x + 2\sin y \cos y \\
   &= \sin 2x + \sin 2y
\end{align}
