Solid angle subtended by polar cap at unit sphere center latitude $\phi$ is
$$ 2 \pi (1- \sin \phi_c)$$
What is the solid angle it subtends at other unsymmetric points inside the sphere like above ... as a function of radius , latitude and longitude: $(r<1,\phi, \theta)$?
On
The formulae for the generic case are in https://vixra.org/abs/1403.0977 . From this it seems no elliptic integrals are needed.
From inside the sphere, outside the spherical cap, the spherical cap subtends the same solid angle as the circular disc at the base of the spherical cap. (You can prove this by examining the rays from the chosen point. If they intersect with the spherical cap, they intersect the circular disk, and vice versa.)
Unfortunately, the solid angle subtended by an arbitrary circular disc involves elliptic integrals (which is kind of obvious, if you think about it). There are a number of nuclear physics papers dealing with calculating this efficiently, as many sensors are circular, and knowing the solid angle the sensor subtends from the sample, is needed to derive the sample properties from the sensor data. For proper mathematical examination, see e.g. Paxton, F. Solid Angle Calculation for a Circular Disk, Review of Scientific Instruments 30 (1959), pp. 254-258 (PDF at UMich).