Simplify the expression $\sin 4x+\sin 8x+\cdots+\sin 4nx$

Question

Simplify the expression $\sin 4x+\sin 8x+\cdots+\sin 4nx$

I have no idea how to do it.

Answer 1

If we multiply the whole sum by $2\sin(2x)$, by exploiting: $$2\sin(4mx)\sin(2x) = \cos((4m-2)x)-\cos((4m+2)x) $$ we have, through telescopic sums: $$ 2\sin(2x)\cdot\sum_{m=1}^{n}\sin(4mx) = \cos(2x)-\cos((4n+2)x).$$

Answer 2

Alternatively,

$\sum_{k=1}^n \sin(4kx) = \Im\left\{\sum_{k=1}^n \exp(4kxi)\right\} = \Im\left\{\frac{1-\exp(4(n+1)xi)}{1-\exp(4xi)}\right\} = \Im\left\{\frac{(1-\exp(4(n+1)xi)\exp(-2xi))}{\exp(-2xi)-\exp(2xi))}\right\} = \frac{\cos(2x)(1-\cos(4(n+1)x))-\sin(2x)\sin(4(n+1)x)}{2\sin(2x)}$