[ Trigonometric Identities. ]

It is known that $\operatorname{tg} \alpha+\operatorname{tg} \beta=p$, $\operatorname{ctg} \alpha+\operatorname{ctg} \beta=q$. Find $\operatorname{tg}(\alpha+\beta)$.
$q=\frac{1}{\operatorname{tg} \alpha}+\frac{1}{\operatorname{tg} \beta}=\frac{\operatorname{tg} \alpha+\operatorname{tg} \beta}{\operatorname{tg} \alpha \operatorname{tg} \beta}=\frac{p}{\operatorname{tg} \alpha \operatorname{tg} \beta}$, hence $\operatorname{tg} \alpha \operatorname{tg} \beta=p / q$. Therefore,

$\operatorname{tg}(\alpha+\beta)=\frac{\operatorname{tg} \alpha+\operatorname{tg} \beta}{1-\operatorname{tg} \alpha \operatorname{tg} \beta}=\frac{p}{1-\frac{p}{q}}=\frac{p q}{q-p}$

## Answer

$p q / q-p$.