For the wallboard joints at the inside corners, you will use a corner trowel tool; there is no need for a mud pan. Just dip the corner trowel right into the joint compound bucket and coat the corner joint well.
Step 1) Start at the top of the wall and press the compound into the corner, holding the blade at a low angle to the wall. Work your way down to about the center, then start at the bottom and work your way up. Working this way lets you avoid ending up with a big glob of compound on the floor. You want to get the compound pressed into the nailhead dimples and the gap between the adjacent drywall sheets.