Let a line makes an angle $\theta$ with $y-,z-$axes and $\varphi$  with $x-$axis. If  $\sin \varphi$ and $\sin \theta$are in the ratio$m:n$ , then $\cos \theta =$

Given  $\varphi ,\theta ,\theta$are the angles made by a line with axes, it implies that 

 $\begin{array}{c}{\cos }^2\varphi +{\cos }^2\theta +{\cos }^2\theta =1\\ 2{\cos }^2\theta =1-{\cos }^2\varphi \\ ={\sin }^2\varphi \end{array}$           

Given  $\frac{\sin \varphi }{\sin \theta }=\frac{m}{n}\Rightarrow \sin \varphi =\frac{m}{n}\sin \theta$

Substitute  $\sin \varphi =\frac{m}{n}\sin \theta$ in the equation  $2{\cos }^2\theta ={\sin }^2\varphi$

It implies that  

$\begin{array}{c}2{\cos }^2\theta =\frac{m^2}{n^2}{\sin }^2\theta \\ {\tan }^2\theta =\frac{2n^2}{m^2}\end{array}$

Use right triangle, we get  $\cos \theta =\boxed{\frac{m}{\sqrt{2n^2+m^2}}}$