10. Given $\alpha+\beta=\frac{\pi}{4},(1+\tan \alpha)(1+\tan \beta)=$
10. 2 .

Given $\alpha+\beta=\frac{\pi}{4}$, thus $\tan (\alpha+\beta)=1$, which means $1-\tan \alpha \tan \beta=\tan \alpha+\tan \beta$, or $\tan \alpha+\tan \beta+\tan \alpha \tan \beta=1$. Therefore, $(1+\tan \alpha)(1+\tan \beta)=1+\tan \alpha+\tan \beta+\tan \alpha \tan \beta=2$.