If A − B = π/4, Find (1 + tan A)(1 − tan B)

Solution

Using the tangent subtraction formula:

\[ \tan(A-B) = \frac{\tan A-\tan B} {1+\tan A\tan B} \]

Given,

\[ A-B=\frac{\pi}{4} \]

Therefore,

\[ \tan\left(\frac{\pi}{4}\right)=1 \]

Hence,

\[ \frac{\tan A-\tan B} {1+\tan A\tan B} =1 \]

Cross multiplying,

\[ \tan A-\tan B = 1+\tan A\tan B \]

Adding \(1+\tan A\tan B\) to both sides,

\[ 1+\tan A-\tan B-\tan A\tan B = 2 \]

Factorizing,

\[ (1+\tan A)(1-\tan B)=2 \]

Therefore,

\[ \boxed{2} \]