drywall detached in corner
It looks as though one or both sides have not been adequately fastened to a framing member or "nailer" (nailer being a wood board that is not a framing member but instead was placed specifically to facilitate the fastening of the wallboard).

You could check by gently pushing against each side along the cracked corner. If you push back and the wallboard stops up against something solid then you may have something to nail/screw to; if not then it was improperly installed without something to nail/screw to.

Tape and mud alone will not work because the wallboard will continue to move and will crack again at that spot.

The best repair option (some may say the only option) is to cut back to the next stud and remove the wallboard, allowing you to install a nailer. Then replace the wallboard and mud/sand/mud/sand ad nauseum...

I had the same situation in my own home, as an experiment I installed 4 small steel corner braces along the crack. I traced their footprint and cut and scraped the wallboard so they sat almost flush, using short drywall screws to anchor them; this held the two panels together at the corner. Then I mudded over them and sanded and primed and painted and it still looks good today (3 years later), no further cracking. This is definitely not the preferred repair method but hey, it might be worth a try if you don't want to tear out the walls.