A line makes the same angle $\theta $, with each of the x and z-axes. If the angle $\bet $, which it makes with y-axis, is such that $sin^2 \beta = 3 sin^2 \theta , $ then $cos^2 \theta $ equals

$\frac{3}{5}$

Let $l, m, n $  be the direction cos (lines. Then, 

$l = cos \theta  , m = cos \beta $ and $ n = cos \theta $

$⇒ cos^2 \theta cos^2 \beta + cos^2 \theta = 1 $

$⇒ 2cos^2 \theta + 1 -sin^2 \beta = 1 ⇒ 2 cos^2 \theta - sin^2 \beta = 0 $

$⇒ 2 cos^2 \theta - 3 sin^2 \theta = 0 $         $[sin^2 \beta = 3 sin^2 \theta $(Given)]

$⇒  tan^2 \theta = \frac{2}{3}$

$∴ cos^2\theta = \frac{1}{1+tan^2\theta }=\frac{1}{1+2/3}=\frac{3}{5}$