Answer
Please see below.
Work Step by Step
We verify as follows:
$$\tan (\alpha + \beta)\tan (\alpha - \ beta )=\frac{\tan \alpha + \tan \beta }{1-\tan \alpha \tan \beta} \frac{\tan \alpha - \tan \beta }{1+\tan \alpha \tan \beta}=\frac{\tan ^2 \alpha - \tan ^2 \beta }{1-\tan ^2 \alpha \tan ^2 \beta}$$
