Thanks everybody, i was pretty sure you couldn't just ignore a force.
I've found a much more elegant way of presenting the solution for the second teacher's method. It's the same up till the part where the expression derived for the normal force is substituted in, and instead of doing that...
I recently ran into something between two teachers. For the longest time one has been teaching that when a car is traveling in circular motion on a banked track with friction, only the horizontal component of the friction force is to be taken into account, by adding to the centripetal force...