Question

If $A+B=\frac{\pi }{4}$, then find the numerical value of $(1+\tan A)(1+\tan B)$.

Answer 1

$A+B=\frac{\pi }{4}\Rightarrow \tan (A+B)=\tan \frac{\pi }{4}=1$
$\Rightarrow \frac{\tan A+\tan B}{1-\tan A+\tan B}=1$
$\Rightarrow \tan A+\tan B=1-\tan A\tan B$
$\Rightarrow \tan A+\tan B+\tan A\tan B=1$.....(i)
Now
$(1+\tan A)(1+\tan B)=1+\tan A+\tan B+\tan A+\tan B=1+1=2$[from(i)]