1. Calculate the sum $S=S_{1}+S_{2}$, where $S_{1}$ and $S_{2}$ are given by

$$
S_{1}=\frac{1}{\log _{\operatorname{tg} 1^{\circ}} 2}+\frac{2}{\log _{\operatorname{tg} 2^{\circ}} 2^{2}}+\ldots+\frac{44}{\log _{\operatorname{tg} 44^{\circ}} 2^{44}} \text { and } S_{2}=\frac{46}{\log _{\operatorname{tg} 46^{\circ}} 2^{46}}+\frac{47}{\log _{\operatorname{tg} 47^{\circ}} 2^{47}}+\ldots+\frac{89}{\log _{\operatorname{tg} 89^{\circ}}{ }^{89}}
$$
Solution. Using the property of logarithms $\log _{x} a^{k}=k \log _{x} a$, the sum

$$
\begin{aligned}
S & =S_{1}+S_{2} \\
& =\frac{1}{\log _{\operatorname{tg} 1^{\circ}} 2}+\frac{2}{\log _{\operatorname{tg} 2^{\circ}} 2^{2}}+\ldots+\frac{44}{\log _{\operatorname{tg} 44^{\circ}} 2^{44}}+\frac{46}{\log _{\operatorname{tg} 46^{\circ}} 2^{46}}+\frac{47}{\log _{\operatorname{tg} 47^{\circ}} 2^{47}}+\ldots+\frac{89}{\log _{\operatorname{tg} 89^{\circ}} 2^{89}} \\
& =\frac{1}{\log _{\operatorname{tg} 1^{1}} 2}+\frac{1}{\log _{\operatorname{tg} 2^{\circ}} 2}+\ldots+\frac{1}{\log _{\operatorname{tg} 44^{\circ}} 2}+\frac{1}{\log _{\operatorname{tg} 46^{\circ}} 2}+\frac{1}{\log _{\operatorname{tg} 47^{\circ}} 2}+\ldots+\frac{1}{\log _{\operatorname{tg} 89^{\circ}} 2}
\end{aligned}
$$

Using the property $\frac{1}{\log _{x} a}=\log _{a} x$, we transform the sum into the form

$$
S=\log _{2} \operatorname{tg} 1^{\circ}+\ldots+\log _{2} \operatorname{tg} 44^{\circ}+\log _{2} \operatorname{tg} 46^{\circ}+\ldots+\log _{2} \operatorname{tg} 89^{\circ}
$$

We will group the terms and using the properties of logarithms and the functions $\operatorname{tg} x$ and $\operatorname{ctg} x$ we get

$$
\begin{aligned}
S & =\log _{2} \operatorname{tg} 1^{\circ}+\log _{2} \operatorname{tg} 89^{\circ}+\log _{2} \operatorname{tg} 2^{\circ}+\log _{2} \operatorname{tg} 88^{\circ}+\ldots+\log _{2} \operatorname{tg} 44^{\circ}+\log _{2} \operatorname{tg} 46^{\circ} \\
& =\log _{2}\left(\operatorname{tg} 1^{\circ} \cdot \operatorname{tg} 89^{\circ}\right)+\ldots+\log _{2}\left(\operatorname{tg} 44^{\circ} \cdot \operatorname{tg} 46^{\circ}\right) \\
& =\log _{2}\left(\operatorname{tg} 1^{\circ} \cdot \operatorname{ctg} 1^{\circ}\right)+\ldots+\log _{2}\left(\operatorname{tg} 44^{\circ} \cdot \operatorname{ctg} 44^{\circ}\right) \\
& =44 \log _{2} 1=44 \cdot 0=0
\end{aligned}
$$