Daisy reveals her cynicism to Nick early in the book, during the events of the first chapter. After Tom leaves dinner to answer a call from his mistress, Daisy says that she's feeling "cynical about everything." She recalls the birth of her daughter, an event at which Tom was nowhere to be found. She says that she hoped her daughter would grow up to be a "beautiful fool," as that's the best thing a woman can be.

Daisy is indeed incredibly cynical about the society in which she operates and her station as a woman. She says that she hopes her daughter is a "fool" because she knows that having insight will not improve her daughter's life in any way, as it has not hers. Knowledge will only cause her to become too aware of the reality of being a woman: the reality that one can only improve their life by marrying a wealthy man.

The reason that Daisy cannot ultimately love Gatsby is because of his unrealistic and ultimately misogynistic view of her. Gatsby states that he wanted her to wait and pine for him during his absence, revealing that for all his charm, he still sees her as nothing more than a prize. She chooses Tom because, for all his misogyny and blatant infidelity, he is at least realistic and grounded in what he is.

In chapter one, Daisy characterizes herself as cynical. She tells Nick:

She hesitated. “Well, I’ve had a very bad time, Nick, and I’m pretty cynical about everything.”

Evidently she had reason to be. I waited but she didn’t say any more

Daisy says this after reminding Nick that he did not attend her wedding to Tom Buchanan. Nick reminds her this is because he was not yet back from the war. Daisy does not explain why she is cynical, but this example of her questioning Nick's reason for avoiding the wedding is just the first example.

Nick changes the subject to Daisy's daughter, and so we see another example of Daisy's cynicism:

It’ll show you how I’ve gotten to feel about—things. Well, she was less than an hour old and Tom was God knows where. I woke up out of the ether with an utterly abandoned feeling

Daisy discusses gender roles, and how she hopes her daughter will be a "beautiful little fool." We see that Tom was not with her during the birth, leaving Daisy to feel alone. Perhaps Daisy is cynical because men have a freedom she does not feel she has.

This first chapter introduces us to Daisy and her cynicism. We see examples of this later in the novel, such as when Nick invites her to his house alone, and Daisy questions, "Are you in love with me?" She is assuming the worst in Nick, or hoping for something to gossip about.

Cynicism is defined by the Merriam Webster dictionary as "beliefs that people are generally selfish and dishonest."

Daisy, for good reason, believes her husband Tom to be dishonest.  She is aware that he is unfaithful and a liar.  In chapter six, when Daisy and Tom attend Gatsby's party, Tom tells Daisy he's going off to eat with another group of people because a man with them is telling funny stories.  Daisy responds, "Go ahead, and if you want to take down any addresses, here's my little gold pencil." Moments later, she tells Nick "the girl was 'common but pretty,'" indicating that she believes Tom went off with another woman at the party.

In chapter seven, Tom takes a phone call in another room, raising his voice and scolding someone (presumably George Wilson) for bothering him at lunchtime about the sale of his car.  In the next room, Daisy says "Holding down the receiver," "cynically" to Nick and the others.  She believes Tom is on the phone with his mistress. 

Later in chapter seven, Daisy calls Tom out for his selfishness and deception, telling him "You're revolting," and implying that they'd left Chicago because of a scandal he'd been involved in, as a reply to his insistence that he "love[s] her all the time."

In The Great Gatsby, one example of Daisy's cynicism is evident when she tells Nick the narrator about the birth of her daughter.

She explains how dissappointed she was when the baby turned out to be a girl instead of a boy.  She tells Nick that she mentally gave in to her daughter's fate.  Very well, then, she says, her daughter must grow up to be a beautiful little fool.  That's the only hope for a female. 

The idea is that Daisy, and her daughter, are women in a man's world.  Women had only recently even received the right to vote.  Divorce laws were heavily weighted in favor of males.  Education was rare for females.  Daisy suggests that the only way for a female to get ahead in American life at the time is to marry wealthy, to be a beautiful little fool, look pretty, do what a man tells her, and marry wealthy.  The American Dream is limited only to men.  The only way women can take part, according to Daisy, is to marry a wealthy man. 

That's what Daisy had to do, of course, and she regrets that her daughter will have to do the same.

Of course, only some wouldcall this cynicism.  Others would say it's realism.