Tape flat angle brackets around the perimeter of the tile, over the surrounding grout. The bracket will protect the tile from any slips by the angle grinder.
Cut around the perimeter of the tile with a utility knife to break the seal of any finish.
Turn the angle grinder on. Hold the angle with both hands and position it so that the wheel spins away from the corner you're cutting toward. Cut around the perimeter of the tile. Cut only as deep as the tile is thick to avoid cutting into the sub floor.
Make one diagonal cut from one corner of the tile to the other. Again, the blade must spin away from the corners you're cutting toward. Make a second diagonal cut to connect the other two corners and create four distinct quadrants in the tile.
Pry up the tile pieces. Place the blade of the putty knife underneath the tile. Tap on the handle of the putty knife with a hammer to pop the tile up. Use the same method to scrape up the old mastic.
Remove the debris from the hole. Clear the dust away with the vacuum's hose attachment.
Clean up any residual mastic with a rag moistened with mineral spirits.
Install fresh mastic and a new tile.