Question

If A − B = π/4, then (1 + tan A) (1 − tan B) is equal to

(a) 2
(b) 1
(c) 0
(d) 3

Answer 1

(a) $2$
$$\begin{gathered} \tan (A-B)=\tan \frac{\mathrm{\pi }}{4} \\ \Rightarrow \frac{\tan A-\tan B}{1+\tan A\tan B}=1 \\ \Rightarrow \tan A-\tan B=1+\tan A\tan B...\left(1\right) \\ \mathrm{Now}, \\ (1+\tan A)(1-\tan B)=1+\tan A-\tan B-\tan A\tan B \\ =1+1+\tan A\tan B-\tan A\tan B\left(\mathrm{Using}\mathrm{eq}\left(1\right)\right) \\ =2 \end{gathered}$$