Question

If $A+B={45}^{\circ },$ then $(1+\tan A)(1+\tan B)=k$. Find the value of $k$.

Answer 1

$A+B={45}^0\Rightarrow \tan (A+B)=\tan \left({45}^0\right)$
$\Rightarrow \frac{\tan A+\tan B}{1-\tan A\tan B}=1$
$\Rightarrow \tan A+\tan B=1-\tan A\tan B\Rightarrow 1+\tan A\tan B+\tan A+\tan B=2\Rightarrow \left(\tan A+1\right)\left(\tan B+1\right)=2\therefore k=2$