Okay, looks I made a few errors---it is a well-known fact engineers cannot calculate things properly ;-). First of all, Blobrana's method of Pyhtagoras-ing the angles and then applying the sine rule is simple and valid, of course. Second---and much more serious---I made an error in the conversion formula: in the expression for P_y the cos(g) should of course read sin(g). (It was very nice to enter a few numbers which mysteriously did not lie a distance 1 away from the origin in Cartesian coordinates, even though that is what I put them at in spherical ones...) Apologies all around. Third correction is that in order to use the astronomical values as given by Blobrana, you need to convert the value of f as well: negate the value before use. Astronomers use a left-handed system *sigh*. Also nice to track down.
So, to give a number example to calculate the angle subtended the hard way: Betelgeuse and Rigel are reported at 88.75 RA, 7 DE; and 77.5 RA, -8 DE; convert that to -88.75, 83; and -77.5, 98 degrees and plug those values into the corrected formulas. You end up with Cartesian coordinates (0.02165, -0.9923, 0.1218) for Betelgeuse and (0.2143, -0.9668, -0.1392) for Rigel---note that for now we do not need to know their distances to the Sun, as we're interested in the angle only, and that is independent of the distance. We can assume those to be 1 for convenience. Vector algebra will then tell you that the inner product of these two vectors equals the cosine of the angle subtended (times the length of both vectors, which is 1, hurray), and if I do the math, I end up with an angle of 18.7 degrees. Which is exactly equal to what you end up with following Blobrana's Pythagoras-approach.
Now I know why we are not taught spherical geometry :-D.
