Proving that $ \tan 2x \cdot (1 + \tan x) \cdot \cot x = \frac{2}{1 - \tan(x)} $ 
Given the following expression,
  $$ \tan(2x) \cdot (1 + \tan(x)) \cdot \cot(x) $$
  the exercise asks to simplify the expression and
  $$ \frac{2}{1 - \tan(x)} $$
  should be the simplified expression.

I have tried everything I possibly could, including letting WolframAlpha eat it to show alternative forms of the expression – nothing worked. 
What do you think? How could I go about simplifying this expression? Thank you.

Defining $t := \tan x$ to save typing ...
$$\begin{align}
\tan 2x (1+\tan x) \cot x = \frac{2t}{1-t^2}\cdot(1+t)\cdot\frac{1}{t} = \frac{2t}{(1+t)(1-t)}\cdot(1+t)\cdot\frac{1}{t} = \frac{2}{1-t} 
\end{align}$$
