MD-MO 2007 Problem 4

The function $f: \mathbb{R}\rightarrow\mathbb{R}$ satisfies $f(\textrm{cot}x)=\sin2x+\cos2x$ , for any $x\in(0,\pi)$ . Find the minimum and maximum value of $g: [-1;1]\rightarrow\mathbb{R}$ , $g(x)=f(x)\cdot f(1-x)$ .