How does adding resistance to a wound rotor induction motor increase torque?
Adding resistance to a wound rotor induction motor does not reduce the maximum torque. The torque at a given slip decreases. The phase angle difference may be reduced slightly because the ratio of rotor reactance to rotor resistance is reduced slightly. I don't think you can attribute the change in the torque vs. speed curve to just the change in the relationship between the rotor resistance and the rotor reactance. You need to look at the stator resistance and reactance as well as demonstrated by the torque equation. See below.


Images from Fitzgerald, Kingsley, Umans, Electric Machinery, 4th ed