Question

If $\theta +\varphi =\pi /4$, then the value of $(1+\tan \theta )/(1+\tan \varphi )$ is .........

Answer 1

Ans. $2$.
We have $\pi /4=\theta +\varphi \Rightarrow \tan \left(\pi /4\right)=\tan (\theta +\varphi )$
$\Rightarrow 1=(\tan \theta +\tan \varphi )/(1-\tan \theta \tan \varphi )$
$\Rightarrow \tan \theta +\tan \varphi +\tan \theta \tan \varphi =1$
$\Rightarrow (1+\tan \theta )(1+\tan \varphi )=2$.