<s>[INST] 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.
 [/INST] \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}
</s>