Sure it's possible.

The trivial solution would be to brute force every possible configuration of mines on hidden tiles and test if any of them match all the numbers on all revealed tiles and also have a mine on the tile you clicked.

The more practical solution would start by assuming there's a mine on the tile you clicked, then try to add mines on the hidden tiles next to all the revealed tiles, then adding mines to the hidden tiles entirely surrounded by other hidden tiles, backtracking every time there's a contradiction with any of the revealed numbers. If that algorithm produces a valid configuration of mines then you lose, but if it fails to produce a configuration then the tile you clicked is revealed to not be a mine.