Since the line makes angle $\theta,\alpha\:and\:\theta$ with the coordinate axes,
$cos^2\theta+cos^2\alpha+cos^2\theta=1$
$\Rightarrow\:2cos^2\theta+cos^2\alpha=1$....(i)
Given: $ sin^2\alpha=2sin^2\theta$
$\Rightarrow\:1-cos^2\alpha=2(1-cos^2\theta)$
$\Rightarrow\:cos^2 \alpha=2cos^2\theta-1$.....(ii)
Solving (i) and (ii), $cos^2\theta=\large\frac{1}{2}$
$\Rightarrow\:cot^2\theta=1$