i am also having trouble with this identity!
tan ( pi /4 + x ) - tan ( pi /4 - x ) = 2 sinx cosx
tan ( pi /4 + x ) + tan ( pi /4 - x )
EDITTED!
The first step is to realize that,
$\displaystyle \tan (\pi/4 + x) = \frac{\tan \pi/4 + \tan x}{1 - \tan \pi/4 \cdot \tan x} = \frac{1+\tan x}{1-\tan x}$
Similarly,
$\displaystyle \tan (\pi/4 - x) = \frac{1-\tan x}{1+\tan x}$.
Now, simplify the fraction and see what you get.