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.
In Act 1V of the play Macbeth by William Shakespeare, the author shows how Malcolm tries to prove that he is trustworthy by speaking truthfully about his own strengths and weaknesses. In particular he says
"Nay, had I power, I should pour the sweet milk of concord into hell..."
Macduff says that this would be a terrible thing for Scotland, and Malcolm casts doubt upon his own ability to govern. Macduff appears to be impressed by his honesty, but disappointed all the same, for his country's future, his miserable nation. Malcolm describes MacBeth as devilish - it appears that Malcolm has only been testing Macduff. Now he believes he can trust him, but fears for his country's safety.
Macduff proves that he is trustworthy by standing up to Malcolm instead of just agreeing with Malcolm or telling him that he is great. This shows Malcolm that Macduff is interested in what's good for Scotland, not in getting ahead himself.
Malcolm tells Macduff that he (Malcolm) will be a terrible king. He says that he is selfish and greedy and things like that. He says it over and over until finally Macduff believes him and says that this is a terrible thing for Scotland.
Once Macduff has done that, Malcolm believes he is a good man.