In Act IV of "Macbeth," how does Macduff prove to Malcolm that he is trustworthy?
In Shakespeare's Macbeth, in, I believe, the longest scene of the play (Act 4.3), Malcolm creates an elaborate test to prove or disprove Macduff's loyalty. In an echo of the fair is foul and foul is fair theme in the play, Malcolm suspects Macduff may be just acting like a friend, when he is really an enemy seeking a chance to betray Malcolm to Macbeth.
The scene is also dramatically ironic, since the audience knows Macduff's family has just been killed on orders by Macbeth, but Malcolm and Macduff don't know this yet.
Malcolm plays a role or puts on an act--another theme in the play--and pretends to be three things: lustful, greedy, and an all-around terrible person. Malcolm is waiting to see if Macduff will continue to support him and go along with him, even when Malcolm convinces him that he'll be a terrible king. If Macduff just blindly keeps saying "Yes, yes, it's okay, we can make this work (I'm paraphrasing)," then Malcolm will assume that Macduff doesn't really care about Scotaland and that he is really an agent for Macbeth.
Macduff goes along with the lust and the greed, saying Scotland has plenty of wenches and plenty of money, but when he hears Malcolm's final test, and Malcolm asks Macduff if he still thinks Malcolm is fit to rule, Macduff replies:
...Fit to govern?
No, not [fit] to live! (Act 4.3.103-104)
And that's enough for Malcolm. He knows Macduff is a loyal Scot and he reveals the truth to Macduff. They form the alliance that will eventually cause Macbeth's destruction.
In another incidence of irony, these two get the news near the end of the scene that Macbeth has had Macduff's family murdered, ending all possible suspicion that Macduff is an agent of Macbeth's.