How can I calculate $\displaystyle\sup_{x\in [-1,1]}\{ |\sin(nx)- \sin(mx)| : m,n\in \mathbb{N}\}$ ?
This is what i have tried
$\sin(nx)- \sin(mx)= 2\cos (\frac{nx+mx}{2})\sin(\frac{nx-mx}{2})$
$\Rightarrow$
$|\sin(nx)- \sin(mx)|= 2|\cos (\frac{nx+mx}{2})||\sin(\frac{nx-mx}{2})|$
But I do not know what else to do...