Question

If $y=(1+\tan A)(1-\tan B)$ where $A-B=\frac{\pi }{4}$, then $(y+1{)}^{y+1}$

Answer 1

$A-B=\frac{\pi }{4}\Rightarrow \tan (A-B)=\tan \frac{\pi }{4}$
$\Rightarrow \frac{\tan A-\tan B}{1+\tan A\tan B}=1$
$\Rightarrow \tan A-\tan B-\tan A\tan B=1$
$\Rightarrow \tan A-\tan B-\tan A\tan B+1=2$
$\Rightarrow (1+\tan A)(1-\tan B)=2\Rightarrow y=2\therefore (y+1{)}^{y+1}=3^3=27$