If d is the diameter of the incident beam with an angle of incidence i on a boundary separating rarer medium and denser medium. If r is the angle of refraction and $d^1$ is the diameter of the refracted beam. Then

Given d is the diameter of the incident beam. 

Let D be the diameter of the profile made by the incident beam on the boundary surface. 

$\mathrm{D}=\frac{\mathrm{d}}{\mathrm{cosi}}$

Similarly on the denser side, 

the profile diameter D is

$\mathrm{D}=\frac{{\mathrm{d}}^1}{\mathrm{cosr}}$

$\Rightarrow \frac{\mathrm{cosi}}{\mathrm{cosr}}=\frac{\mathrm{d}}{{\mathrm{d}}^1}$