160204 初版 160204 更新
\(\tan\(\alpha+\beta)\)
\(=\dfrac{\sin(\alpha+\beta)}{\cos(\alpha+\beta)}\)
\(=\dfrac{\sin\alpha\cos\beta+\cos\alpha\sin\beta}{\cos\alpha\cos\beta-\sin\alpha\sin\beta}\)
\(=\dfrac{\frac{\sin\alpha}}{\cos\alpha}+\frac{\sin\beta}{\cos\beta}}
{1-\frac{\sin\alpha}{\cos\alpha}\cdot\frac{\sin\beta}{\cos\beta}}\)
\(=\dfrac{\tan\alpha+\tan\beta}
{1-\tan\alpha\tan\beta}\)