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

$$
(1+\operatorname{tg} \alpha)(1+\operatorname{tg} \beta)=\left(1+\frac{\sin \alpha}{\cos \alpha}\right)\left(1+\frac{\sin \beta}{\cos \beta}\right)=\frac{\cos \alpha+\sin \alpha}{\cos \alpha} \times
$$

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

Answer: 2.