Vtyjkyuk

This seems like it should be fairly simple, but it has me completely stumped. Imagine I have a latitude line at angle $theta_1$ on the surface of the unit sphere in 3D. This is a "small circle", meaning it's a circle that's not a great circle.

Now suppose I have, in addition, a second coordinate system where the axis is at an angle $phi$ to the original axis, and a second latitude line at an angle $theta_2$ in the new coordinate system:

If these two curves cross, they do so at two points. I want to know the angle they make on the surface of the sphere. That is, the angle between the vectors tangent to the two circles at a point where they cross:

What is this angle $psi$, as a function of $phi$, $theta_1$ and $theta_2$?

The answer is simple once you see the spherical geometry of the situation. Suppose you have a point $P$ on the sphere. There are two other points $P_1,P_2$ each at fixed angular distances $Theta_1,Theta_2$ from $P$. The two "latitude" angles $theta_1,theta_2$ are the respective complementary angles. The two points $P_1,P_2$ are at an angular distance $phi$ from each other. The three points $P,P_1,P_2$ form a spherical triangle. Now the small circle centered at $P_1$ and passing through $P$ has a tangent line on the sphere surface which is perpendicular to the great circle determined by $P,P_1$ which forms one side of the spherical triangle. Similarly with the small circle centered a $P_2$. Thus, the angle $psi$ between the two tangent lines is the same as the angle at $P$ of the spherical triangle. This angle can be found using the spherical law of cosines since the angular distance between the three points $P,P_1,P_2$ are known. The cosine law states
$$ cos phi = cos Theta_1 cos Theta_2 +
sin Theta_1 sin Theta_2 cos psi $$
and since $Theta_1,Theta_2$ are complementary to $theta_1,theta_2$ we get
$$ cos psi = (cos phi - sin theta_1 sin theta_2 ) / (cos theta_1 cos theta_2). $$