For my cosmology assignment, we're required to derive an expression for the source count $\frac{dN}{dS}$ and show that it is $$ \cfrac{dN}{dS} ∝ \cfrac{S^{-3}}{\cos \theta}$$
We start from a Universe that resides on the surface of a sphere and we consider a spherical and an observer at the pole. I was able to show that the volume enclosed using polar coordinates was $$V=2\pi R^2(1-\cos \theta)$$, and that the flux for the observer is $$ S= \cfrac{L}{2 \pi R \sin \theta}$$
Where I am confused is that we are asked to find an expression for $$ \cfrac{dV}{d \theta}$$ and $$\cfrac{dN}{d\theta} $$ and I am not too sure how to go on about this as all substitutions I have made like $$\sin \theta ∝ s^{-1}$$ do not seem to lead to the expected result.
I came up with $$dN = \text{density} \times \text{surface area}\, d\theta$$
and
$$dV ∝ \cos \theta d \theta$$
since we are talking about a 2d universe.
A slight nudge in the right direction will be greatly appreciated!
