One can use vector analysis to find the points where two great circle cross each other. This is an alternative to using spherical trigonometry or a search for a solution. A great circle on a unit sphere can be specified by a point it passes through and its tangent at that point. A third unit vector at the point and perpendicular to the first two unit vectors is the binormal and can be found by taking the cross product of the unit vectors for the point and its tangent. The binormal points in the direction of the axis of rotation that would move a point on the great circle along it. It is the same everywhere on the great circle.
The distance of an arbitrary point on the sphere from the great circle is the length of the shortest arc connecting them. This arc is part of another great circle that is perpendicular to the first and can be defined by their common point and the direction of the binormal of the first great circle. To get to the arbitrary point one travels along two arcs, the first along the great circle and a second perpendicular to it. One can use this procedure to find the perpendicular distance of a point on a second great circle from the first great circle. Where the they meet this distance is zero.
To set up a problem I chose two arbitrary points on the equator of a unit sphere and arbitrary directions for the intersecting great circles and then computed the vectors for the points and the tangents. The formula for the unknown angle is found by taking the dot product of the initial position vector and the tangent for the second great circle with the binormal of the first. Once the angle is found the intersection point can be found by a simple rotation along the great circle.
A plot shows that the two great circles do indeed meet at this point.
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