I've cranked through the math, and can now greatly simplify the solution.
Imagine if we spin the earth so that our first vector is at 0 degrees latitude and 0 degrees longitude. The second vector would be at (alpha2 - alpha1) degrees latitude and (beta2 - beta1) degrees latitude.
Since ...
sin(0) = 0 and cos(0) = 1
our dot product simplies to ...
cos(delta_alpha) * cos(delta_beta) = cos(theta)
The rest of the math remains unchanged.
theta = acos (cos(delta_alpha) * cos(delta_beta))
distance = radius * theta
Hope this helps.
