$\begin{array}{l}\text { 3. If } \operatorname{tg} x+\operatorname{tg} y=25, \text { and } \\ \quad \operatorname{ctg} x+\operatorname{ctg} y=30, \\ \text { find } \quad \operatorname{tg}(x+y) .\end{array}$
$\begin{array}{l}\text { 3. From ctg } x+\operatorname{ctg} y=30 \text { we get } \frac{1}{\operatorname{tg} x}+\frac{1}{\operatorname{tg} y} \\ =30 \Rightarrow \operatorname{tg} x+\operatorname{tg} y=30 \operatorname{tg} x \cdot \operatorname{tg} y . \\ \text { That is } \operatorname{tg} x \cdot \operatorname{tg} y=\frac{\operatorname{tg} x+\operatorname{tg} y}{30} \\ =\frac{25}{30}=\frac{5}{6} \text {. } \\ \text { Therefore, }(x+y)=\frac{\operatorname{tg} x+\operatorname{tg} y}{1-\operatorname{tg} x \operatorname{tg} y} \\ =\frac{25}{1-\frac{5}{6}}=15 \text {. } \\\end{array}$