Find a point of tangency in a plane from a point to a sphere using spherical coordinates

Question

I'm facing a problem very similar to this one, but I can't resolve mine: find ANY point of tangency from a point to a sphere using spherical coordinates

Here's the situation:

Illustration

I know the spherical coordinates of $B$ and $D$. I know the radius of the sphere $S$. I have the $P$ plane equation in the form $(ax + by + cz + d) = 0$ (that I found by computing vector product of $OB$ and $OD$)

Now I want to get the coordinates of $C$ (which I believe is the unique point of tangency in the plane P). It's similar to the question I've linked because instead of finding ALL points, I want to find THE point that is in the plane $P$.

Any help would be very much appreciated.