4. Find the sum

$$
\log \operatorname{tg} 1^{\circ}+\log \operatorname{tg} 2^{\circ}+\ldots+\log \operatorname{tg} 89^{\circ}
$$
Solution. We have

$$
\begin{aligned}
\log \operatorname{tg} 1^{\circ} & +\log \operatorname{tg} 2^{\circ}+\ldots+\log \operatorname{tg} 89^{\circ}=\log \left(\operatorname{tg} 1^{\circ} \cdot \operatorname{tg} 2^{\circ} \cdot \ldots \cdot \operatorname{tg} 89^{\circ}\right) \\
& =\log \left(\operatorname{tg} 1^{\circ} \cdot \operatorname{tg} 2^{\circ} \cdot \ldots \cdot \operatorname{tg} 44^{\circ} \cdot \operatorname{tg} 45^{\circ} \cdot \operatorname{ctg} 44^{\circ} \cdot \ldots \cdot \operatorname{ctg} 1^{\circ}\right) \\
& =\log \left(\left(\operatorname{tg} 1^{\circ} \cdot \operatorname{ctg} 1^{\circ}\right) \cdot\left(\operatorname{tg} 2^{\circ} \cdots \operatorname{ctg} 2^{\circ}\right) \ldots \cdot\left(\operatorname{tg} 44^{\circ} \cdot \operatorname{ctg} 44^{\circ}\right) \cdot \operatorname{tg} 45^{\circ}\right) \\
& =\log (1 \cdot 1 \cdot 1 \cdot \ldots \cdot 1)=\log 1=0
\end{aligned}
$$

## 4th year