Put down a plastic tarp or a painter's cloth underneath the area where you will be working. This will be a timesaver when cleaning up.
Cut along both sides of the corner crack with a utility knife, about a 1/4-inch from the edges.
Scrape, but do not gouge, the area between the knife cuts you made to loosen and remove any debris. Use a paint scraper or flat-head screwdriver for this action.
Cut a piece of drywall tape or mesh that measures the length of the crack.
Place a very thin layer of drywall mud with your drywall knife over the scraped area in the corner. Put the tape or mesh on top of the wet compound.
Put another thin layer of drywall mud on top of the tape. The thinner the better, as any excess will have to be sanded later.
Allow the mud to dry completely.
Spread a thin layer of mud on only one side of the corner, as even with the wall as possible. When dry, spread a thin layer of mud on the other side, using the tip of your drywall knife to make a clean line in the corner.
Sand any excess material to make the walls and corner smooth.