Origin of the neutron diffusion angle formula

[Nafis Fuad]

I found this formula when I was studying neutron diffusion equation in Weston M. Stacey's book
Thanks in advance

cosβ=cosθx cosθ+sinθx sinθ cos(ψy−ψ)

Edited by moderator for clarity:
$\cos(\beta) = \cos(\theta_x)\cos(\theta) + \sin(\theta_x)\sin(\theta)\cos(\psi y - \psi)$

 

[mfb]

How is the second line related to the first one?
Where do these variables come from and how are they related? The equation is not true in general.

 

[Nafis Fuad]

Here it is

 

[mfb]

$\psi - \psi_y$ is the angle between the two planes of (one vector and x) and (the other vector and x).

We can imagine $\beta$, $\theta$ and $\theta_r$ (as drawn) to form a triangle on a sphere with radius 1, with $\psi - \psi_y$ as angle between the last two. The problem is then standard geometry on a sphere, see e.g. Wikipedia.