Question

If $A+B=\frac{\mathrm{\pi }}{4},$ then (1 + tan A) (1 + tan B) = ___________.

Answer 1

$$\begin{gathered} \mathrm{Since}A+B=\frac{\mathrm{\pi }}{4} \\ \left(1+\tan A\right)\left(1+\tan B\right)=? \\ \tan \left(A+B\right)=\tan \frac{\mathrm{\pi }}{4} \\ \mathrm{i}.\mathrm{e}.\frac{\tan A+\tan B}{1-\tan A\tan B}=1 \\ \mathrm{i}.\mathrm{e}.\tan A+\tan B=1-\tan A\tan B \\ \tan A+\tan B+\tan A\tan B=1 \\ \mathrm{i}.\mathrm{e}.\tan A+1+\tan B\left(1+\tan A\right)=1+1 \\ \mathrm{i}.\mathrm{e}.\left(1+\tan A\right)\left(1+\tan B\right)=2 \\ \mathrm{Hence},\mathrm{value}\mathrm{of}\left(1+\tan A\right)\left(1+\tan B\right)=2. \end{gathered}$$