In Hamlet, Gertrude tells the king: Hamlet is "mad as the sea and the wind when both contend/which is the mightier." Does she betray Hamlet?
In Shakespeare's play, Hamlet, I believe Gertrude is keeping her word to Hamlet.
First, early on we know that Hamlet puts on an "antic disposition" to throw those around him off, so they don't know if he is an enemy and dangerous, or simply insane.
When Claudius enters at the beginning of Act IV, scene i, his first question is "Where is your son?" Instead of telling him, or answering immediately, Gertrude excuses Rosencrantz and Guildenstern, and replies that she has seen terrible things.
Claudius then asks how Hamlet is? We know the King cares nothing for his step-son. We can only assume that he is trying to ascertain what kind of a danger Hamlet is.
Gertrude protects Hamlet by continuing to preserve the myth he has built up around himself, that he is mad. Her description presents images of a man out of control, whose sanity has fled.
To know that Gertrude is in earnest of protecting her son, return to the previous act. At the end of Act III, when Hamlet has killed the man hidden behind the curtain, Gertrude asks what he has done. Hamlet's response is very telling to Gertrude:
Nay, I know not. / Is it the King? (III.iv.29-30)
Hamlet thinks he has killed Claudius (finally). It is now that Hamlet describes the betrayal of Claudius and the murder of his father. Gertrude is stunned. Hamlet forces his mother to look inside herself, and then to perceive what Claudius has done and the kind of man she has married.
Gertrude has been told the truth of her second husband. When Hamlet is ready to condemn her, Old Hamlet's ghost appears and tells Hamlet to leave her punishment to heaven. Though she has not seen the ghost of Old Hamlet, her son has as they stand together. Seeing Hamlet transfixed by his father's spirit persuades Gertrude. It as if a veil has been lifted from her eyes so that she sees Claudius for who he truly is, for the first time since their hasty marriage. It is a moment of true self-enlightenment for Gertrude.
Hamlet implores Gertrude (though her duty as Queen might compel her to do otherwise) not to tell Claudius that Hamlet is not mad. She responds with a solemn promise—I will not tell him that you are sane, that your behavior is a pretense.
QUEEN:Be thou assur'd, if words be made of breath,
And breath of life, I have no life to breathe
What thou hast said to me. (III.iv.214-216)
And so, when Gertrude reports to Claudius that Hamlet is mad, she is protecting him and his confidence to her as she lies to her husband. Gertrude personifies the elements of nature; she says Hamlet is as mad as the sea and wind when they battle to see who is mightier— elements enraged and out of control:Mad as the sea and wind when both contend
Which is the mightier. (IV.i.7-8)