Find spherical coordinates from which to define great circle I've found a formula for defining a great circle (since it's the set of points $(\theta, \varphi)$ such that their distance is $\pi/2$ from a given point $(\theta_0, \varphi_0)$):
$-\tan(\varphi)\tan(\varphi_0) = \cos(\theta_0 - \theta).$
Now, I have two points on the sphere $(\theta_1,\varphi_1),(\theta_2,\varphi_2)$. I want to use them to define a great circle, but to do that, I need to use those two points in some way to derive $(\theta_0, \varphi_0)$ (either one of the two possibilities assuming the two points aren't antipodal or equal). How can I do so (whether using the above formula or another part of spherical geometry)?
Note: I want to do this entirely with spherical geometry and spherical coordinates if possible. I know that there's a very easy way to do it: simply take the cross product of the 3-dimensional vector representation of the two points and normalize it. But I'm also doing a lot of other things in spherical coordinates and so hopefully a spherical geometry solution will help me with other similar problems.
 A: If $\varphi_1=\pm\frac\pi2,$ if $\varphi_1=\pm\frac\pi2,$
or if $\theta_1 - \theta_2$ is an integer multiple of $\pi$
then you know $\varphi_0=0$ and you can easily find $\theta_0$ if it is determined.
If $\varphi_1=0$ (or if $\varphi_2=0$) then you know that 
$\theta_0 = \theta_1 \pm \frac\pi2$
(or $\theta_0 = \theta_2 \pm \frac\pi2,$ respectively)
and can then easily find $\varphi_0$ if it is determined.
Now let's assume a case in which none of the conditions above is true.
This implies that $\tan\varphi_0\neq 0.$
You know that if your great circle's pole is at $(\theta_0,\varphi_0)$ then your two points must satisfy the equations
\begin{align}
-\tan\varphi_1\tan\varphi_0 &= \cos(\theta_0 - \theta_1)
= \cos\theta_0\cos\theta_1 + \sin\theta_0\sin\theta_1,  \tag1\\
-\tan\varphi_2\tan\varphi_0 &= \cos(\theta_0 - \theta_2)
= \cos\theta_0\cos\theta_2 + \sin\theta_0\sin\theta_2. \tag2
\end{align}
Cross-multiply Equation $(1)$ and Equation $(2)$ to get
$$
-\tan\varphi_1\tan\varphi_0 (\cos\theta_0\cos\theta_2 + \sin\theta_0\sin\theta_2)
= -\tan\varphi_2\tan\varphi_0 (\cos\theta_0\cos\theta_1 + \sin\theta_0\sin\theta_1).
$$
Since $\tan\varphi_0\neq 0,$ divide by $-\tan\varphi_0$ on both sides and distribute the multiplication to get
$$
\tan\varphi_1 \cos\theta_0\cos\theta_2 + \tan\varphi_1 \sin\theta_0\sin\theta_2
= \tan\varphi_2 \cos\theta_0\cos\theta_1 + \tan\varphi_2 \sin\theta_0\sin\theta_1.
$$
Collect terms in $\cos_0$ and $\sin_0$:
$$
(\tan\varphi_1 \cos\theta_2 - \tan\varphi_2 \cos\theta_1) \cos\theta_0
= (\tan\varphi_2 \sin\theta_1 - \tan\varphi_1 \sin\theta_2) \sin\theta_0.
$$
Since $\varphi_1,$ $\varphi_2,$ $\theta_1,$ and $\theta_2$ are all known, 
you now have an equation of the form
$k_1 \cos\theta_0 = k_2 \sin\theta_0$ for known $k_1$ and $k_2$ and you can solve for $\theta_0.$
For example, if $\tan\varphi_2 \sin\theta_1 - \tan\varphi_1 \sin\theta_2 \neq 0$
then you can set
$$
\theta_0 =
\arctan\left(\frac{\tan\varphi_1 \cos\theta_2 - \tan\varphi_2 \cos\theta_1}
                  {\tan\varphi_2 \sin\theta_1 - \tan\varphi_1 \sin\theta_2}\right).
$$
Once you know $\theta_0$ you can use it to solve for $\varphi_0$ in one of equations $(1)$ or $(2).$
