3.177. Calculate $(1+\operatorname{ctg} \alpha)(1+\operatorname{ctg} \beta)$, if $\alpha+\beta=\frac{3 \pi}{4}$.
Solution.

$$
\begin{aligned}
& (1+\operatorname{ctg} \alpha)(1+\operatorname{ctg} \beta)=\left(1+\frac{\cos \alpha}{\sin \alpha}\right)\left(1+\frac{\cos \beta}{\sin \beta}\right)=\frac{\sin \alpha+\cos \alpha}{\sin \alpha} \times \\
& \times \frac{\sin \beta+\cos \beta}{\sin \beta}=\frac{\cos \alpha \cos \beta+\sin \alpha \sin \beta+\sin \alpha \cos \beta+\cos \alpha \sin \beta}{\sin \alpha \sin \beta}= \\
& =\frac{\cos (\alpha-\beta)+\sin (\alpha+\beta)}{\frac{1}{2}(\cos (\alpha-\beta)-\cos (\alpha+\beta))}=\frac{2\left(\cos (\alpha-\beta)+\frac{\sqrt{2}}{2}\right)}{\cos (\alpha-\beta)+\frac{\sqrt{2}}{2}}=2
\end{aligned}
$$

Answer: 2.