This can be confusing. Latitude lines run EW but are spaced in NS distances and on the right of the chart. Longtitude lines run NS but give EW angles. The distance between them at the equator is the biggest and at the poles zero. So stand on the pole and turn around and you do a 360 in longtitude, but your latitude stays the same.
What you are doing is trying to change your longtitude at the pole by 1 minute 1/360 of arc true but not a mile - only a mile at the equator. There is a formula that gives changes in longtitude as you near the poles. Someone once used the table to test my knowledge or rather more to establish superiority. I had shelved it as irrelevant as I had no intention of sailing near the poles and we were in the tropics at the time.