\(sin\left(\alpha+2\beta\right)=3sin\alpha\)
\(\Leftrightarrow sin\left(\alpha+\beta\right).cos\beta+cos\left(\alpha+\beta\right).sin\beta=3sin\left(\alpha+\beta-\beta\right)\)
\(\Leftrightarrow sin\left(\alpha+\beta\right).cos\beta+cos\left(\alpha+\beta\right).sin\beta=3\left[sin\left(\alpha+\beta\right).cos\beta-cos\left(\alpha+\beta\right).sin\beta\right]\)
\(\Leftrightarrow4.cos\left(\alpha+\beta\right).sin\beta=2.sin\left(\alpha+\beta\right).cos\beta\)
\(\Leftrightarrow2.\dfrac{sin\beta}{cos\beta}=\dfrac{sin\left(\alpha+\beta\right)}{cos\left(\alpha+\beta\right)}\)
\(\Leftrightarrow tan\left(\alpha+\beta\right)=2.tan\beta\)
