加法定理の証明

$3$\displaystyle{\tan $ {\alpha}+{\beta} $ }$

$3$\displaystyle{\begin{array}{lll}\tan $ {\alpha}+{\beta} $ & \vspace{6}\\ & \vspace{6}\\ \text{\hspace{5}}=\frac{\hspace{2} \sin $ {\alpha}+{\beta} $ \hspace{2}}{\hspace{2} \cos $ {\alpha}+{\beta} $ \hspace{2}}& \vspace{6}\\ \end{array}}$

$3$\displaystyle{\text{\hspace{5}}=\frac{\hspace{2} \sin {\alpha}\cos {\beta}+\cos {\alpha}\sin {\beta} \hspace{2}}{\hspace{2} \cos {\alpha}\cos {\beta}- \sin {\alpha}\sin {\beta} \hspace{2}}}$

ここで，
分子，分母を $3$\displaystyle{\cos {\alpha}\cos {\beta}}$ で割ると，

$3$\displaystyle{\text{\hspace{5}}=\frac{\hspace{2} \tan {\alpha}+\tan {\beta} \hspace{2}}{\hspace{2} 1- \tan {\alpha}\tan {\beta} \hspace{2}}}$

となる。

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