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:

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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
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Which is the mightier.
(IV.i.7-8)