Chapter 7 - Trigonometric Identities and Equations - Summary Exercises Verifying Trigonometric Identities - Exercises - Page 696: 23

$$\dfrac {\tan \left( x+y\right) -\tan y}{1+\tan \left( x+y\right) \tan y}=\tan \left( x+y-y\right) =\tan x$$

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