Thousands of sums

Let

$S_n=sin(x) - sin (2x)+...+(-1)^{n-1} sin(nx)$ and $C_n=cos(x)-sin(2x)+...+(-1)^{n-1}cos(nx)$

$\Sigma=C_n+i S_n=(cos(x) + i sin(x))-(cos(2x)+i sin(2x))+...$

Now let $q=cos x + i sin x$

$\Sigma=q-q^2+q^3-...=q(1-q+q^2-q^3+...)=q(1-(\Sigma-(-1)^{n-1} q^n))$

You will easy find that:

$\Sigma=\frac{q(1+(-1)^{n-1}q^n)}{q+1}$

By using trigonometric identities you can rewrite sigma as:

$\Sigma=\frac{cos(\frac{x}{2})+(-1)^{n-1}cos(\frac{x}{2})cos(nx)-(-1)^{n-1} sin(\frac{x}{2}) sin(nx)}{2cos(\frac{x}{2})} +i \frac{sin(\frac{x}{2})+(-1)^{n-1}cos(\frac{x}{2})sin(nx)+(-1)^{n-1} sin(\frac{x}{2}) cos(nx)}{2cos(\frac{x}{2})}$

It's clear that:

$C_n=\frac{cos(\frac{x}{2})+(-1)^{n-1}cos(\frac{x}{2})cos(nx)-(-1)^{n-1} sin(\frac{x}{2}) sin(nx)}{2cos(\frac{x}{2})}$

And

$S_n=\frac{sin(\frac{x}{2})+(-1)^{n-1}cos(\frac{x}{2})sin(nx)+(-1)^{n-1} sin(\frac{x}{2}) cos(nx)}{2cos(\frac{x}{2})}$

For odd $n$ $S_n$ can be written as:

$S_n=\frac{cos(\frac{nx}{2}) sin(\frac{(n+1)x}{2})}{cos(\frac{x}{2})}$