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.

We have better inequality Since $|\cos a|\le 1$ and 
$$ \sqrt 2 (\cos(x+\frac{\pi}{4})) =\sqrt 2 (\cos x\cos\frac\pi4-\sin x\sin\frac\pi4)  =\sin x -\cos y .$$
we have, 
$$\color{blue}{|\sin x -\cos y|  =\sqrt 2|\cos(x+\frac{\pi}{4})|\le\sqrt2<2.}$$
