How to proof : $\sin(\pi k\Delta x)+\sin(\pi k3\Delta x)=2\cos(\pi k\Delta x)\sin(\pi k 2\Delta x$)

Question

How to proof: $\sin(\pi k\Delta x)+\sin(\pi k3\Delta x)=2\cos(\pi k\Delta x)\sin(\pi k 2\Delta x)$. Tried all the trig identities but it doesn't seem to work.

Answer 1

Use,$$\sin A + \sin B = 2\cos\frac{A-B}{2}\sin\frac{A+B}{2}$$

And $$\cos(-x) = \cos(x)$$