Given
$3\tan (\theta-15^{\large\circ})=\tan (\theta+15^{\large\circ})$
$\large\frac{\tan (\theta+15^{\large\circ})}{\tan (\theta-15^{\large\circ})}=\frac{3}{1}$
By componendo & dividendo
$\large\frac{\tan (\theta+15^{\large\circ})+\tan (\theta-15^{\large\circ})}{\tan (\theta+15^{\large\circ})+\tan (\theta-15^{\large\circ})}=\frac{3+1}{3-1}$
$\Rightarrow \large\frac{4}{2}$$=2$
$\large\frac{\Large\frac{\sin (\theta+15^{\large\circ})}{\cos (\theta+15^{\large\circ})}+\Large\frac{\sin (\theta-15^{\large\circ})}{\cos (\theta-15^{\large\circ})}}{\Large\frac{\sin (\theta+15^{\large\circ})}{\cos (\theta+15^{\large\circ})}-{\Large\frac{\sin (\theta-15^{\large\circ})}{\cos (\theta-15^{\large\circ})}}}=$$2$
$\large\frac{\sin(\theta+15^{\large\circ})\cos(\theta-15^{\large\circ})+\sin(\theta-15^{\large\circ})\cos(\theta+15^{\large\circ})}{\sin(\theta+15^{\large\circ})\cos(\theta-15^{\large\circ})-\sin(\theta-15^{\large\circ})\cos(\theta+15^{\large\circ})}$$=2$
$\large\frac{\sin(\theta+15^{\large\circ}+\theta-15^{\large\circ})}{\sin(\theta+15^{\large\circ}-\theta+15^{\large\circ})}$$=2$
$\Rightarrow \large\frac{\sin 2\theta}{\sin 3\theta}=$$2$
$\Rightarrow 2\sin 2\theta=2$
$\sin 2\theta=1=\sin\large\frac{\pi}{2}$
$\therefore 2\theta=n\pi+(-1)^n\large\frac{\pi}{2}$
$\therefore \theta=\large\frac{n\pi}{2}$$+(-1)^n\large\frac{\pi}{4}$
Hence (A) is the correct answer.