Why are yaw rate and pitch rate obtained by these equations?
Turn rate is referenced to earth axes, yaw rate to aircraft axes, hence the $cos\Phi$:


If you turn at $\Phi$ = 0, your rate of turn is the yaw rate and the pitch rate is zero. 
If you turn at $\Phi$ = 90°, your yaw rate is zero and your pitch rate = rate of turn.


In order to maintain a steady turn and not lose altitude the pilot does need to pull on the stick when turning..