Question

If $A+B=\frac{\pi }{4}$, find $(1+\tan A)(1+\tan B)$.

• A. $1$
• B. $\frac{\sqrt{3}}{2}$
• C. $0$
• D. $2$

Answer 1

The correct option is D $2$

Given, $A+B=\frac{\pi }{4}$

$\tan (A+B)=\tan \frac{\pi }{4}$

$\frac{\tan A+\tan B}{1-\tan A\tan B}=1$

$\tan A+\tan B=1-\tan A\tan B$

$\tan A+\tan B+\tan A\tan B=1$

$1+\tan A+\tan B+\tan A\tan B=2$

$(1+\tan A)+\tan B(1+\tan A)=2$

$(1+\tan A)(1+\tan B)=2$