Although Shakespeare represents Desdemona as an overwhelmingly virtuous character, he also shows us that she is a flawed human being like everyone else, and can act unwisely. This is true of her in two main contexts: her impetuous elopement with Othello, revealed in Act 1, scene 1; and her dishonest response to Othello's request to see the handkerchief in Act 3, scene 4.
The truth of the matter is that Desdemona has accidentally dropped the handkerchief in Act 3, scene 3 when mopping Othello's brow, in perfectly innocent circumstances. As usual, her motives are pure and, in this instance a sign of love and concern. In response to Othello's "I have a pain upon my forehead, here" (3.3.288), Desdemona replies "Let me bind it hard..." (3.3.290). Othello, however, expresses an impatient desire to go inside and, following him, she drops the handkerchief without noticing it fall to the ground.
By the time the two meet again, Othello's mind has been so thoroughly poisoned by Iago, that he is ready to attack Desdemona for the slightest sign of any indiscretion or unusual behaviour. Specifically, Iago has informed Othello in the interim that he has seen Cassio wipe his beard (3.3.440-442) with a handkerchief “spotted with strawberries” (3.3.438). Othello sets Desdemona up by claiming to have a cold: “I have a salt and sullen rheum offends me” (3.4.51). He asks her to lend him her handkerchief, but, unable to do so, she replies, “I have it not about me” (3.4.55). Her main fault consists in not explaining that she only very recently lost it which, of course, is something that can happen to anyone. We must remember that at one level, it is only a handkerchief, but that, at another, it is to become for Othello the “ocular proof” (3.3.365) of Desdemona’s infidelity. Clearly, Desdemona knows that Othello is out of sorts (“My Lord is not my Lord”: 3.4.125), but this is a very unwise first move. Othello is quick to pounce (‘That’s a fault.”: 3.4.57), and then proceeds to expound on the magical properties of the handkerchief, and its significance as insurance against infidelity.
Desdemona then compounds her first mistake by making three further inopportune comments. The first of these is in response to Othello’s revelation that the handkerchief was sewn by a prophetess from the silk of sacred worms. Her nervous and clumsy outburst, “Then would to God that I had never seen’t!” (3.4.79) infuriates Othello and adds fuel to the fire of his anger. The second compounding fault is to lie to Othello by claiming that she will be able to produce the handkerchief later: “Why so I can, sir: but I will not now” (3.4.82). This is followed immediately by her claiming light-heartedly that Othello’s over-reaction to the missing handkerchief is a trick to prevent her from pleading Cassio’s case (3.4.89-90). The genius of Shakespeare allows the light-hearted response of Desdemona to become entangled once again with Othello’s monstrous jealousy, and his now intense hatred of Cassio. Othello demands to have the handkerchief in three more rapid-fire outbursts, all of which Desdemona meets with an innocent but misjudged reference to Cassio. Othello exits enraged.