If A − B = π/4, Find (1 + tan A)(1 − tan B)
Question:
If \[ A-B=\frac{\pi}{4} \] then \[ (1+\tan A)(1-\tan B) \] is equal to
If \[ A-B=\frac{\pi}{4} \] then \[ (1+\tan A)(1-\tan B) \] is equal to
Solution
Using the identity:
\[ \tan(A-B) = \frac{\tan A-\tan B} {1+\tan A\tan B} \]
Given,
\[ A-B=\frac{\pi}{4} \]
Therefore,
\[ \tan(A-B)=\tan\frac{\pi}{4}=1 \]
Hence,
\[ \frac{\tan A-\tan B} {1+\tan A\tan B} =1 \]
Cross multiplying,
\[ \tan A-\tan B = 1+\tan A\tan B \]
Rearranging,
\[ 1+\tan A-\tan B-\tan A\tan B = 2 \]
Factorizing,
\[ (1+\tan A)(1-\tan B)=2 \]
Final Answer
\[ \boxed{ (1+\tan A)(1-\tan B)=2 } \]
Correct Option: (a)