Let a line makes an angle ‘$\theta$’ with  $X$ and $Z$ -axes and $\beta$ with $Y$ -axis. If  $\sin \beta =\sqrt{3}\sin \theta ,$$10{\cos }^2\theta +2=$_________ 

${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$

$\therefore {\cos }^2\theta +{\cos }^2\theta +{\cos }^2\beta =1$

$2{\cos }^2\theta ={\sin }^2\beta \mathrm{....}\left(1\right)$

$\sin \left(\beta \right)=\sqrt{3}\sin \theta \left(given\right)$

${\sin }^2\beta =3{\sin }^2\theta \mathrm{......}\left(2\right)$

$\therefore 2{\cos }^2\theta =3{\sin }^2\theta$

$\Rightarrow 3(1-{\cos }^2\theta ),5{\cos }^2\theta =3,{\cos }^2\theta =\frac{3}{5}$