詳解：複角公式

\(\tan (\alpha  + \beta ) = \frac{{\tan \alpha  + \tan \beta }}{{1 - \tan \alpha \tan \beta }} = 1\)

\(\frac{\frac{1}{9}+ \tan \beta}{1- \frac{1}{9}\tan \beta} = 1 \Rightarrow \frac{1+9\tan\beta}{9-\tan\beta}=1\)

\(1 + 9\tan \beta  = 9 - \tan \beta \)

\(10\tan \beta  = 8\quad  \Rightarrow \tan \beta  = \frac{8}{{10}} = \frac{4}{5}\)