How does Prospero tame Caliban in The Tempest?

Prospero teaches Caliban basic skills and then continually threatens him. 

Caliban believes the island is his by right, not Propsero’s. Prospero has magic, though, and can make the inhabitants of the island obey his will. He is particularly cruel with Caliban because he believes Caliban tried to assault his daughter Miranda. Prospero uses Caliban as a slave and punishes him relentlessly. 

When Prospero first landed on the island, he seems to have had a...

Prospero teaches Caliban basic skills and then continually threatens him. 

Caliban believes the island is his by right, not Propsero’s. Prospero has magic, though, and can make the inhabitants of the island obey his will. He is particularly cruel with Caliban because he believes Caliban tried to assault his daughter Miranda. Prospero uses Caliban as a slave and punishes him relentlessly. 

When Prospero first landed on the island, he seems to have had a better relationship with Caliban. Caliban showed him where to find food and fresh water, and Prospero and Miranda taught Caliban their language and other features of their culture, such as the Man in the Moon. Then Caliban targeted Miranda, and that was it. 

PROSPERO

Thou most lying slave,
Whom stripes may move, not kindness! I have used thee,
Filth as thou art, with human care, and lodged thee
In mine own cell, till thou didst seek to violate
The honour of my child. (Act 1, Scene 2) 

Caliban doesn’t deny it, he just says he wishes he was successful because he would have “peopled else/This isle with Calibans.” As a result of this conflict, Prospero calls Caliban his slave, and constantly threatens him with violence.

PROSPERO

If thou neglect'st or dost unwillingly
What I command, I'll rack thee with old cramps,
Fill all thy bones with aches, make thee roar
That beasts shall tremble at thy din.

CALIBAN

No, pray thee.

Aside

I must obey: his art is of such power,
It would control my dam's god, Setebos,
and make a vassal of him. (Act I, Scene 2) 

Caliban knows that, as long as Prospero has magic, there is little he can do. That’s why Caliban tries to convince Trinculo and Stephano to kill Prospero. The three bumble through the rest of the play until Prospero finally forgives Caliban when he gives up his magic and leaves the island.