A line makes the same angle $\theta$, with each of the x and z-axes. If the angle $\beta$, which it makes with y-axis, is such that

${\sin }^2\beta =3{\sin }^2\theta \text{, then }{\cos }^2\theta$ equals

Let $l, m, n$ be the direction cosines of a line Then,

$\begin{array}{l}l=\cos \theta ,m=\cos \beta \text{ and }n=\cos \theta \\ \therefore l^2+m^2+n^2=1\\ \Rightarrow {\cos }^2\theta +{\cos }^2\beta +{\cos }^2\theta =1\\ \Rightarrow 2{\cos }^2\theta +1-{\sin }^2\beta =1\Rightarrow 2{\cos }^2\theta -{\sin }^2\beta =0\\ \Rightarrow 2{\cos }^2\theta -3{\sin }^2\theta =0\left[{\sin }^2\beta =3{\sin }^2\theta \left(\text{ Given }\right)\right]\\ \Rightarrow {\tan }^2\theta =\frac{2}{3}\\ \therefore {\cos }^2\theta =\frac{1}{1+{\tan }^2\theta }=\frac{1}{1+2/3}=\frac{3}{5}\end{array}$