A) \[9\]
B) \[4\]
C) \[27\]
D) \[81\]
Correct Answer: C
Solution :
Given, \[A-B=\frac{\pi }{4}\] \[\Rightarrow \] \[\tan (A-B)=\tan \frac{\pi }{4}\] \[\Rightarrow \] \[\frac{\tan A-tanB}{1+\tan \,A\,\tan {{B}^{-1}}}\] \[\Rightarrow \] \[\tan A-\tan B-\tan A\,\tan B=1\] ??(i) Also given, \[y=(1+\tan A)(1-\tan B)\] \[=(1-tanB+\tan A-\tan A\,\tan B)\] \[=(1+1)\] [from Eq. (i)] \[=2\] \[\therefore \] \[{{(y+1)}^{y+1}}={{(2+1)}^{2+1}}={{3}^{3}}=27\]
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