A) 9
B) 4
C) 27
D) 81
Correct Answer: C
Solution :
\[A-B=\frac{\pi }{4}\,\Rightarrow \,\tan \,(A-B)=\tan \frac{\pi }{4}\] \[\Rightarrow \,\,\frac{\tan A-\tan B}{1+\tan A\,\tan B}=1\] \[\Rightarrow \,\,\tan A-\tan B-\tan A\,\tan B=1\] \[\Rightarrow \,\,\tan A-\tan B-\tan A\,\tan B+1=2\] \[\Rightarrow \,\,(1+\,\tan A)\,\,(1-\tan B)=2\] Þ \[y=2\] Hence, \[{{(y+1)}^{y+1}}={{(2+1)}^{2+1}}={{(3)}^{3}}=27\]. Trick : Put suitable A and B as \[A-B=\frac{\pi }{4}\] i.e.,\[A=\frac{\pi }{4},B=0\] \[\therefore \,\,\,\left( 1+\tan \frac{\pi }{4} \right)\,(1-\tan {{0}^{o}})=2(1)=2\].
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