Answer
$$\dfrac {\tan \left( x+y\right) -\tan y}{1+\tan \left( x+y\right) \tan y}=\tan \left( x+y-y\right) =\tan x$$
Work Step by Step
$\tan \left( \alpha -\beta \right) =\dfrac {\tan \alpha -\tan \beta }{1+\tan \alpha \tan \beta }\Rightarrow $
$$\dfrac {\tan \left( x+y\right) -\tan y}{1+\tan \left( x+y\right) \tan y}=\tan \left( x+y-y\right) =\tan x$$