Explanation

(b)
$\tan{\theta + \tan{\left( \frac{3\pi}{4} + \theta \right) = 2}}$
$\Rightarrow \ \tan\theta + \tan{\left( \frac{\pi}{2} + \left( \frac{\pi}{4} + \theta \right) \right) = 2}$
$\Rightarrow \ \ \tan{\theta - \cot{\left( \frac{\pi}{4} + \theta \right) = 2}}$
$\Rightarrow \ \ \tan{\theta - \frac{\cot{\frac{\pi}{4}\cot{\theta - 1}}}{\cot{\frac{\pi}{4} + \cot\theta}}} = 2$
$\Rightarrow \ \tan{\theta - \frac{\cot{\theta - 1}}{1 + \cot\theta} = 2}$
$\Rightarrow \ \tan{\theta - \frac{1 - \tan\theta}{1 + \tan\theta} = 2}$
$\Rightarrow \ \ \tan{\theta + \tan^{2}{\theta - 1 + \tan{\theta = 2 + 2\tan\theta}}}$
$\Rightarrow \ \tan^{2}{\theta = 3}$
$\Rightarrow \ \ \tan{\theta = \pm \sqrt{3} = \pm \tan\frac{\pi}{3}}$
$\Rightarrow \ \ \theta = n\pi \pm \frac{\pi}{3},\ n \in Z$