i'm about to drywall my ceiling in my living room. my stairway is also in this area. at the bottom of the stairway is a joist that makes the stairway's height going under the joist a little less than 6 foot... i have to duck. i was wondering if i could trim this joist about 3-5 inches, in a 3' wide area and drywall a approx 45 degree angle from the 2nd joist past the stairwell to the cut joist, to the end of the stairwell... giving me a couple more inches of head room.

Of course i don't want to compromise the ceiling... can i do this safely?

 

Not enough info. What direction do the joist run relative to the stairs? What loads are above? If you are lucky, the opening may already be constructed by headering off a joist. Making the opening longer might not involve to much structural work. If the stairs run perpendicular to the rafters, it could require a lot more work. You should get at least a carpenter familiar with framing to look over what yo have.