Claudius demands the Queen explain her son; Gertrude claims that the Prince is insane. Does she believe this or just protect him from Claudius?
Given the fact that Gertrude deliberately drinks the poison intended for Hamlet in the final act of the play, it would seem that the queen is being somewhat protective when she tells Claudius that her son is insane. For, she is guilt-ridden after Hamlet's chastisement in Act III of "Hamlet":
This was your husband/....Could you on this fair mountain leave to feed,/And batten on this moor?...You cannot call it love, for at your age/The heyday in the blood is tame, its humble,/And waits upon the judgment, and what judgment/Would step from this to this?/...What devil was't/That thus hath cozened you at hoodman-blind?...O shame where is thy blush? (III,iv,64-83)
O Hamlet, speak no more!/Thou turn'st my eyes into my very soul,/And there I see such black and grained spots/As will not leave their tinct. (III,iv,89-92)
At this point, when the ghost enters and Hamlet speaks to this apparition, Gertrude is moved to remark, "Alas, he's mad," but her remarks are yet tinged with her own guilt,for she asks Hamlet,
Alas, how is't with you,/That you do bend your eye on vacancy,/And with th'incorporal air do hold discourse?/...O gentle son,/Upon the heat and fame of thy distemper/Sprinkle cool patience....Whereon do you look?....This is the very coinage of your brain.
Then, as Hamlet retorts,
Forgive me this my virtue,/For in the fatness of these pursy time/Virtue itself of vice must pardon beg,/Yea, curb and woo for leave to do him good. (III,iv,156-159)
To this vituperation, Gerturude replies, "O Hamlet, thou hast cleft my heart in twain" (III,iv,166). Again, the reader is led to believe that Gertrude blames herself for Hamlet's madness; her "unvirtuous" actions have brought Hamlet to this state. Earlier in Act II Gertrude tells Claudius that Hamlet's "distemper" is due to "His father's death and our o'erhasty marriage." Because of her guilt and earlier defense of Hamlet, the reader may believe that Gertrude hopes to mitigate the consequences for Hamlet's murder of Polonius. Even in her response to Claudius's question as to where Hamlet has gone, she displays her love for Hamlet:
To draw apart the body he hath killed,/O'er whom his very madness, like some ore/Among a mineral of metals base,/Shows itself pure:'a weeps for what is done (IV, i, 24-27)
The short first scene of Act IV centres around Gertrude's betrayal of Hamlet. Although she does honour Hamlet's request to not tell Claudius that he is only feigning madness, it is clear that, having promised to help her son, Gertrude turns him in to her husband. The immediate and extremely frank way in which Gertrude reports the actions of Hamlet in killing Polonius show that she is well and truly on the side of her husband. Beyond this, the motives of Gertrude are up to your own individual opinion: some have argued that she truly believed Hamlet to be mad and thus wanted to help him by reporting him to Claudius, others say that she has recognised that her best interests lie in the camp of Claudius at this stage and this is why she betrays Hamlet. Either way, it cannot be avoided that it is this act of Gertrude that leads to Claudius' plan to have Hamlet killed. "Frailty thy name is woman" indeed!