The most general value of $\theta$ satisfying $\tan \theta +\tan (\frac{3\pi }{4}+\theta )=2$ are

The correct option is A $n\pi \pm \frac{\pi }{3},nϵZ$

Simplifying it,
$\tan \theta +\frac{\tan \theta -1}{\tan \theta +1}=2$

$\tan \theta (1+\tan \theta )+\tan \theta -1=2(1+\tan \theta )$

${\tan }^2\theta =3$

$\tan \theta =\sqrt{3}=\tan \frac{\pi }{3}$

So,
$\theta =n\pi \pm \frac{\pi }{3}$