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:
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.