$$3=\frac{\sin (\alpha +\beta +\beta )}{\sin (\alpha +\beta -\beta )}=\frac{\sin (\alpha +\beta )\cos \beta +\cos (\alpha +\beta )\sin \beta }{\sin (\alpha +\beta )\cos \beta -\cos (\alpha +\beta )\sin \beta }$$，  分子分母同时除以$$\cos \beta \cos (\alpha +\beta )$$，  有$$\frac{\tan (\alpha +\beta )+\tan \beta }{\tan (\alpha +\beta )-\tan \beta }=3$$，解得$$\tan (\alpha +\beta )=2\tan \beta $$． 
已知$$\frac{\sin \left( \alpha +2\beta \right)}{\sin \alpha }=3$$，且$$\beta 
e \frac{1}{2}k \pi, \alpha +\beta 
e n \pi +\frac{ \pi }{2}\left( n,k\in Z \right)$$，则$$\frac{\tan \left( \alpha +\beta \right)}{\tan \beta }$$的值为．

  * $$2$$ 
  * $$1$$ 
  * $$\frac{1}{2}$$ 
  * $$-2$$ 