Why is the torque exerted by the magnetic field in a moving coil galvanometer independent of the angle of rotation? How does an iron core cause the magnetic field to be constant and always perpendicular to the loop's moment in a galvanometer? 
The following image is from my textbook:

Image source: NCERT Physics Textbook for Class XII Part I - Page 164
Put another way, why is the torque independent of the angle?
 A: I hope that you have come across the following formula to determine the torque $\tau$ exerted by a magnetic field on a current carrying loop:
$$\vec\tau=ni(\vec A\times\vec B)$$
where, $n$ is the number of loops in the loop, $i$ is the current it carries, $\vec A$ is the area of the loop and $\vec B$ is the magnetic field.
The pole pieces are made cylindrical as shown in the following diagram:

You could see that the field lines are almost parallel to the plane of the loop. Or in other words, the magnetic field $\vec B$ is perpendicular to the area vector $\vec A$.
So torque exerted by the magnetic field on the coil is:
$$\tau=niAB\sin\theta$$
Put $\theta=90^{\circ}$:
$$\tau=niAB$$
Now coming to your question:

Why is the torque independent of the angle?

The torque is independent of the angle because the magnetic poles are made cylindrical. If the poles were just flat (like an ordinary bar magnet), then the torque obviously depends upon the angle, but the case is different here.
A: The cylindrical core of a galvanometer is mounted within a cylindrical gap between the N and S poles of a permanent magnet. The magnetic field lines cross a small gap between the poles and the core, going toward the center of the core as they enter from the N pole and away from the center as they cross to the S pole.
A: I think I got it. 
The iron core being ferromagnetic cause field lines to go through it when they might not have gone through that area in the core's absence. (This is due to the magnetic field created by the core's magnetisation being added to the magnet's field to make the new net field.)
This causes the following rough pattern of field lines inside the core:
Thus because of the core's ferromagnetism, the lines stay almost radial except at the centre, and at any angle of the loop, the field is radial for the most part, and so the equation used is a slight approximation. However, because of soft iron's strong ferromagnetism, the inaccuracy is tiny, as experiments have shown. 
The key thing which solves the doubt is that two field lines together provide a field parallel to the loop's plane, and one alone cannot, as it change direction in the centre. 
