Question:
If $A+B=\frac{\pi}{4}$, then $(1+\tan A)(1+\tan B)=$
Solution:
Since $A+B=\frac{\pi}{4}$
$(1+\tan A)(1+\tan B)=?$
$\tan (A+B)=\tan \frac{\pi}{4}$
i. e. $\frac{\tan A+\tan B}{1-\tan A \tan B}=1$
i. e. $\tan A+\tan B=1-\tan A \tan B$
$\tan A+\tan B+\tan A \tan B=1$
i. e. $\tan A+1+\tan B(1+\tan A)=1+1$
i. e. $(1+\tan A)(1+\tan B)=2$
Hence, value of $(1+\tan A)(1+\tan B)=2$.
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