Question:
If A − B = π/4, then (1 + tan A) (1 − tan B) is equal to
(a) 2
(b) 1
(c) 0
(d) 3
Solution:
(a) 2
$\tan (A-B)=\tan \frac{\pi}{4}$
$\Rightarrow \frac{\tan A-\tan B}{1+\tan A \tan B}=1$
$\Rightarrow \tan A-\tan B=1+\tan A \tan B \quad \ldots(1)$
Now,
$(1+\tan A)(1-\tan B)=1+\tan A-\tan B-\tan A \tan B$
$=1+1+\tan A \tan B-\tan A \tan B \quad($ Using eq $(1))$
$=2$
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