Show that $f(x)=\cos(x)$ is Lipschitz continuous function. I am not sure how to proceed. Should I rewrite $|\cos(x)-\cos(y)|=|2\cos\frac{x+y}{2}\cos\frac{x-y}{2}|$?

For all $x$, $|\sin x|\leq |x|$. Using Sum to Product formula:
$|\cos(x)-\cos(y)|=\left|-2\sin\left(\dfrac{x+y}{2}\right)\;\sin\left(\dfrac{x-y}{2}\right)\right|\leq2\left|\sin\left(\dfrac{x-y}{2}\right)\right|\leq2\left|\dfrac{x-y}{2}\right|=|x-y|$
