Show that $|\sin x−\cos x|≤ 2$ for all $x.$ Please help me out in answering this:
Show that $|\sin x−\cos x|≤ 2$ for all $x.$
I don't know where to start and I think we have to use the mean value theorem to show this.

The mean value theorem also works, since 
$$
\sin(x+\pi/2)-\sin x=-\frac{\pi}{2}\cos\xi,
$$
where $\xi\in(x,x+\pi/2)$.  It follows that $|\cos x-\sin x|\le \pi/2$.  [This is, of course, not as sharp as achille hui's approach.]
