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

$\frac{3}{5}$

We know that sum of squares of direction ratios is 1

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

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

so $2 \cos ^2 \theta =\sin ^2 \beta$        (given $\sin^2 \beta = 3 \sin^2 \theta$)

$2 \cos ^2 \theta =3 \sin ^2 \theta$ 

so  $2 \cos ^2 \theta =3-3 \cos ^2 \theta$

$\Rightarrow \cos ^2 \theta  =\frac{3}{5}$

Option: 3