angles on the celestial sphere This is a somewhat simplified, and drawn / formulated in a manner more targeted to astronomy, version of a diagram and a question that I've also posted in math.stackexchange.com.

In the diagram below I depict:


the celestial sphere with the North Celestial Pole (NCP), the South Celestial Pole (SCP) and the celestial equator
a latitude circle at latitude $\delta$ (declination)
two points $A$ and $B$ on that latitude circle
the center of the celestial sphere $O$ and the center of the latitude circle $O'$




My question is:

Given angle $\phi$ between the rays $OA$ and $OB$, what is the angle $\theta$ between the rays $O'A$ and $O'B$ ? I am trying to derive $\theta$ given $\phi$ and $\delta$. Observe that if $A$ is the first point of Aries, then the angle $\theta$ is the Right Ascension of a star located at point $B$.

My assumption is that the following formula holds:

$\phi = \theta*cos(\delta)$
This answer in math.stackexchange provides the correct relationship between $\theta$ and $\phi$. And this answer supplies the proof.