They don't cut off tusks. Elephant tusks are teeth and they cannot be cut off. Rhinos have horns, and their horns are made of keratin which is what fingernails are made of and that is why you can cut off Rhino horns.
Elephant tusks being teeth helped create the myth of Cyclops. Tusks, like teeth, can fall out of skulls once the connective tissue is gone. They also break easily compared to the other teeth. They found old elephant skulls that were huge along with leg bones that looked like human legs so they thought they were giants. Elephants have no bone between their eyes, there is just one big hole in the center of their skull for their eyes. This made the ancient populations think that the creature had just one eye. The giant bones and the big hole in the skull made them create the Cyclops.
This may be a replica but it's accurate: