Is there a faster way to integrate this? for the following integration: 
$\displaystyle\int\limits^{{\pi}}_{0} \dfrac{2}{{\pi}}\sin\left(x\right)\cos\left(2nx\right)\,\mathrm{d}x$
can I simplify sin(x)cos(2nx) into a simpler form to make the integration easier/faster? i remember there being a simplification for it, but I also remember that it followed certain conditions, if I'm not mistaken.

Hint:
Use the identity $$\sin \alpha\cos \beta=\frac12\sin(\alpha+\beta)+\frac12\sin(\alpha-\beta)$$
