One of the best types of lies are ones which make people want to stop asking questions altogether. A good way to accomplish this is for the lie to be about something You Don't Want to Know.
An Embarrassing Cover Up is great not only because it can cause the person questioning them to lose interest, but also because they are a good cover-up for suspicious behavior. Bob might be suspicious if Alice claims she snuck off in the middle of the night without telling anyone to get ice cream. If she claims she did so to replenish her tampon supply, on the other hand, there's at least a 9 in 10 chance that Bob will leave it at that.
"Women's problems", in fact, are a very common use of the trope. There is even an example of this happening in biblical scripture, making this trope Older Than Feudalism. For men, explosive diarrhea is a much more popular choice.
Sometimes an Embarrassing Cover Up can create friction in a group that's keeping a secret, particularly if someone makes an embarrassing cover up for someone else. For example:
Bob and Alice are working on Project X in total secret. One day, Carol catches them together and asks what they're doing. To Bob's horror, Alice proceeds to invent a cover story which is hugely embarrassing to Bob. But Bob has to go along with it or risk making Carol suspicious.
This trope overlaps with Infraction Distraction if the cover story is an actual admission of guilt (albeit to a much smaller offense than the one being concealed) in addition to an embarrassing disclosure.
See also Sure, Let's Go with That. For a more literal "embarrassing cover up," see Hand or Object Underwear (or maybe Naked Freak-Out).
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Anime & Manga
InuYasha regularly features Kagome's grandfather inventing embarrassing or improbable explanations for her absence from school while she was cavorting around in the Feudal Era. Most of these were illnesses, getting weirder and weirder (and worse and worse) as the series went on. Kagome lampshades it after the first few times it happens, while her grandfather is telling a friend of hers another embarrassing illness. This was in the earliest episodes. How could it get worse? This included herpes at one point in the dub.
In Axis Powers Hetalia, England was performing a secret black ritual to curse Germany. Then America walks in and England quickly "invites" him to the costume party.
In CLANNAD, Kyou has been riding her "bike" (actually a motorized scooter) to school (which is apparently against the rules), and keeps cutting off Tomoya after he gets out the morae "bai." A friend later asks what he was talking about, and he proclaims, for the entire class, that Kyou is bi.
At one point in My Bride Is a Mermaid, Nagasumi has to help cover up Masa's past... by pretending to be gay and having a secret love for him. The episode ends with Nagasumi attempting to convince his (female) fiance that the whole "I'm gay and in love with Masa" thing was made up.
In IDW's G.I. Joe series, Kwinn and Lighthorse are smuggling Snake-Eyes across a border in a coffin inside a hearse. Lighthorse looks nervous and the border guards ask him what is wrong. Lighthorse responds with the phrase that Kwinn taught him the local language. This translates as "I'm a little girl. I'm afraid of ghosts". The border guards laugh and let them through. Lighthorse, who has no idea what he has just said, wonders what they found so funny.
In the KanonGender Flip fic Canon In G, Kaoru (m!Kaori) tells Yuuko (f!Yuuichi) not to disclose the contents of their conversation about the demons in the school. When pressed by Junri (m!Jun), who's likely to get the wrong idea if she's left with the knowledge that Kaoru's keeping something from her, Yuuko gleefully explains that they were instead discussing Kaoru's "embarrassing rash".
Films — Live-Action
The Santa Clause Scott Calvin's ex-wife is commenting sarcastically on the Santa pajamas he got at the North Pole. As she's driving away with their son, Charlie, he yells back a few comments about how he sleeps, including, "I sleep buck naked!" right as his neighbor walks by.
Superhero Movie has two characters doing it to themselves, in a parody of a scene from Spider-Man. They both note injuries the other has sustained in battle as their respective alter egos. Starts with the fairly innocuous "I burned [my wrist] on some hot coffee." Escalates with "[My lip is cut because] my crack pipe broke." Continues until the villain excuses himself because, "I shat my pants."
In Crash, an LAPD officer wants to be transferred away from his racist partner, but won't go on record explaining why. When he won't reveal his partner's racism on record, his black supervisor, well aware of his true motivations, refuses to allow the transfer unless he explains (officially, and on the record) that he has extremely embarrassing and uncontrollable flatulence, and that he wishes to be assigned to a personal squad-car in order to avoid imposing on a partner. The supervisor also explains that it would negatively affect both of their careers to go on record about the racism so the cover story is their only option.
In The Amazing Spider-Man, Gwen Stacy gets rid of her dad while Peter is in her bedroom by claiming she has "cramps" and further implying they are menstrual in nature.
Ruthless People: Judge Reinhold's character has kidnapped a woman. Some cops come in to question him, and he excuses himself saying he has a touch of stomach flu. They hear moans and groans from the back room and assume he's having serious diarrhea, but he's really trying to sneak out a back window. Meanwhile, the cops have received a call implicating the husband, so they just leave.
Harry has secret lessons with Snape to learn Occlumency. When Malfoy comes upon a lesson, Snape explains that Harry is in remedial Potions.
The Dursleys tell their friends that Harry attends St Brutus' Secure Centre for Incurably Criminal Boys in order to cover up the time he spends at Hogwarts. This is less embarrassing to Harry, however, as he generally cares very little what the Muggle neighbors think of him. (He was still understandably aghast when he first found out about this, though.)
In Incarnations of Immortality, this dynamic is part of Parry's backstory, explaining how he was adopted by a Sorcerer. The Sorcerer and the Lord were planning to use baby Parry as a human sacrifice, but they were caught in the act by the Abbot. The Abbot would not have approved, and would have excommunicated both of them, so the Lord said that the sacrificial altar was just for sacrificing a sheep, and he explained what a baby was doing there by claiming that the Sorcerer was going to adopt this baby. So the Sorcerer had to go along with this and adopt Parry as his own son.
In the Discworld novel Moving Pictures, Victor and Ginger get trapped by a rockslide while Victor is trying to stop Ginger walking in her sleep (and apparently trying to awaken an Eldritch Abomination). The characters who dig them out assume the obvious explanation for them being off in a remote spot together, and Victor agrees that they've got it exactly right; when Ginger objects, he asks her if she'd rather tell them what she was really doing.
How To Be A Superhero has an example of a swamp monster attempting to maintain a Secret Identity and trying to explain away the smell by claiming to have had 'an accident' in his pants.
In Robert A. Heinlein's story "If This Goes On", the protagonist is advised to conceal resistance secrets by using a code that makes them look like notes for an illicit gambling system. Thus, if the notes are discovered, there will be a credible explanation for why they were hidden and the protagonist will draw only minor punishment.
In Wearing Black to the White Party by David Stukas, the protagonist Robert locks himself in Rex's office so he can snoop through Rex's papers in order to find out who has been sending threatening letters. When Rex asks why the office was locked, Robert claims that he was photocopying his butt.
In an episode of Breaking Bad, Walter lies about having explosive diarrhea as an excuse for not helping Hank investigate his boss, who is distributing methamphetamine.
In another case, Skyler tells Hank and Marie that Walter's illicit money came from gambling, rather than admitting he manufactures methamphetamine.
And, of course the classic excuse for his 3-day Meth cook was to end it walking out of the desert naked into a convenience store claiming it was a fugue state, then telling the psychiatrist who was verifying it that he came up with the "fugue state" to cover up running out on his family but thinking better of it.
In a Murphy Brown episode, Murphy helps a fellow employee, Hillary, who she recognizes has a drinking problem. After Hillary realizes she needs help after screwing up an assignment, Murphy suggests using food poisoning as a cover story, as nobody wants to know the details.
Jack invites Liz to a party he wants to be kept secret. After he leaves her office, he tries to cover this by loudly saying "Well Lemon, that was a good chat. (Gestures to his crotch) Good luck with that alopecia problem of yours!" ("I don't have alopecia! I'm very hairy!")
Liz and Pete sneak around to find a new cast member, and eventually, to keep this a secret, they claimed to be in an intimate relationship. Of course, this backfires when just as they say this, Pete's wife shows up.
Chandler tries to stop Monica entering the apartment while he and Ross are trying to fix the furniture by saying "You can't come in! Ross is naked!". This is just as embarrassing for Chandler as it was for Ross, though.
Monica: Why is Ross naked? Ross: Um, a "guy problem". Monica: Oh, is it the same thing Chandler has?
Then there was the arc when Chandler and Monica were seeing each other but didn't want anyone else to know. Except Joey had figured it out (yes, that Joey) and they had sworn him to secrecy. This, of course, meant that he had to cover for their slip-ups, such as when Chandler's underpants were found in Monica's apartment ("Oh, they're mine."). This culminated in Monica and Chandler "outing" Joey as a sex addict to the group, at which point he turns the tables on them by announcing that he had slept with Monica in London and she'd become obsessed with trying to get him back in the sack ever since. Monica's forced to endorse his version, thus keeping the secret, allowing Joey to get a little revenge for the way he'd been taken for granted and - most importantly of all - protecting his reputation as The Casanova.
"So, Buffy, how'd the slaying go last night?", "Xander!", "I mean, how'd the laying go? ...no, I don't mean that either."
"How could you love an umpire? Everybody hates those."
Cheers: In the first episode Sam asks Dianne what he should say if any of his regulars ask why she's hanging around the bar, she says he can tell them whatever he wants. The first time someone asks, Sam says she's a hooker.
Chuck: Chuck occasionally explains away Sarah's absence by blaming it on a spastic colon.
In the White Collar pilot, Peter and Neal need access to a church that is closed for renovations. Neal tells the priest that Peter is on the verge of committing adultery and that a visit to the church where he was married would help Peter save his marriage. They get access.
Malcolm in the Middle, the protagonist has to bring some confort item to a person that secretly lives in Malcolm mother's workplace. When she questions him about having a pineapple, spare pants and a book with him, he explains that he is going through a very bad belly problem, justifing the need of fresh fruit, a book to read in the bathroom and well, spare pants. Self-inflicted trope.
In Psych, Shawn loves to make these at Gus' expense. Oftentimes he'll do it even when telling the truth wouldn't get them in trouble. A favorite tactic of Shawn's is to pretend that he and Gus are gay lovers in situations where revealing their real relationship — Heterosexual Life-Partners — would work just as well.
Darryl claims to have bought "nasty old-lady porn" after purchasing an e-book reader at a bookstore (potentially awkward when one works at a paper company.)
Angela slaps Dwight for being insensitive about her cat's death which he caused and people nearby notice and look concerned, he covers with "It's okay, I'm robbing her."
When Dwight refuses to talk to Pam on the phone (despite sitting opposite her), Nate gives the excuse that "Dwight is being questioned by the police in connection to a string of dognappings".
In Frasier, in the episode "Travels with Martin", Frasier decides to take his Winnebago vacation party to Canada on a whim, which is a problem for Daphne, as she doesn't have her Green Card yet. When they are pulled over trying to cross the border back into the US, there is a tense scene as we wonder whether Daphne's accent will give her away. However, Martin saves the day by inventing a story that the dog hasn't had his shots to explain why they all seem nervous.
In Misfits, when Curtis is busy cheating on Samantha and she turns up at the community centre demanding to know where he is, Simon and Nathan cover for him by saying he's been arrested for "exposing himself" to some Boy Scouts. She doesn't seem to believe them, but she leaves anyway, convinced they're nuts and eager to get as far away from them as possible (particularly when Nathan starts casually spraying her with cleaning fluid.
Attempted by Rudy in the Nazi timeline. He's caught in Curtis's bar during lockdown and tries to explain his presence by claiming to be Curtis's lover. As this is a Nazi timeline, this doesn't work out very well for them.
In an episode of That '70s Show, Eric tries to hide the fact that he was reading Donna's journal by claiming to have been going through her underwear drawer and rubbing her underwear against his face.
The Closer: In "Off the Hook", Provenza is trying to conceal that he is talking to Brenda on the phone:
Pope: Is that Chief Johnson? Provenza: No, it's my proctologist. Pope: You're asking where your proctologist is?
This trope abounds in Merlin, where Arthur constantly believes the most embarrassing excuse possible whenever Merlin can't explain something. (Merlin walking around with two dresses=Merlin cross-dressing in one occasion.) The one that actually became a Running Gag was mentioned in Queen Of Hearts, where Gaius says that Merlin is at the tavern. He's missing for two whole days once and Arthur believes he's been in the tavern the whole time. Is it any wonder he's the show's Master of Delusion?
Heroes invokes the trope when Claire covers for the super of the week by telling her mom that they were having sex in the closet.
In the series premier of BlackAdder, the title character is secretly nursing Henry Tudor back to health in his room, hoping for a reward. His mother comes in, and when Edmund implies that there is someone in his bed, the Queen asks if it is a sheep. He denies this. Later on, to keep her from checking the bed he makes bleating noises. Her response: "Oh, Edmund. It's the lying I find so hurtful."
The Latest Buzz: In "The Pet Peeves Issue", Amanda is attempting to hide her pet pony in the office. When the others complain about the smell of horse dung, Amanda claims it is her new perfume.
In an Rizzoli & Isles episode, Frankie tells Jane that he thinks that Frost's mother is in a secret relationship with her (female) roommate. When Frost suddenly shows up and asks what they are talking about, they say that they are talking about their mother and whether she might be going through menopause.
In the Cabin Pressure episode "Helsinki", Douglas is smuggling contraband orchids to Finland. He claims that he and the man receiving the contraband explored their Situational Sexuality together at Boarding School and that the flowers (and the fish the man gives Douglas in exchange) are presents given in fond memory of their time together.
Jeff Foxworthy advises this in one of his early routines if you have to call in sick to work. Basically, Jeff says to call in with something so gross and disgusting that the boss won't ask questions about.
Jeff Foxworthy: If you're gonna fake an illness, fake something so gross that you know they won't ask too many questions about it - like explosive diarreha. Then they will be like "No, that's okay, Jefferson. In fact, we'll pay you to stay home if you've got that."
Myths & Religion
In The Bible, Jacob and Rachel flee from Rachel's father ,Laban, stealing some of his household idols as they go. Laban catches up to them and demands to search their tents for the idols. Rachel sits on top of them and when Laban gets to her tent and tells her to move, she claims that "the manner of women" is upon her, so she can't get up. Laban believes her and leaves.
Nicky: Yes, definitely, I would say my buddy Rod is a closeted Homosexual. Rod: WHAT?! Nicky: yes, definitely, my buddy Rod has... uh... an undescended testicle.
In the classic farce, No Sex, Please, We're British, Newly married Peter and Frances pull this on their friend Brian constantly throughout the play in order avoid a scandal. The best example would be when Peter tells his and Brian's boss, Mr Bromhead, that Brian is suffering from — something — which is why the two need to talk alone. Bromhead recomends that Brian get some ointment for it as soon as possible.
In Ansem Retort, Aerith falls down and hits a doorknob, which gives her a black eye. Naturally, she blames it on Axel. Who is at first, furious, then quickly covers it up and goes with it. (Also an Even Evil Has Standards moment, since that is the only point Axel isn't outright enthusiastic about an act of violence.) He may be a mass murderer who will kill you sooner than look at you, but dammit he loves his wife.
In Ménage ā 3, when Yuki needed Gary to take a day off, she called his boss, and told her that Gary needs to have a foreign object removed that he inserted into his butt; "who'd make that up?". This eventually got Gary into trouble, when he ended up in a hospital (for completely different reasons) and met his boss there.
In one strip of The Order of the Stick, Haley has a huge sack, and claims she didn't find any treasure. When Roy asks what's in the sack, she replies, "Feminine products." Roy decides to leave it there.
In Girl Genius, Mamma Gkika's has a notorious Jägermonster-themed burlesque show. Mamma also provides discreet medical care to people who cannot openly go to the hospital. Gil is brought there after he is injured while helping Agatha against his father's wishes. When told where Gil is, Klaus just raises the priority of finding Gil a girlfriend.
Insurance Agent: Now this place you were at, Moe's, is this a business of some sort? Homer's Brain: Don't tell him you were at a bar! *gasp* But what else is open at night? Homer: It's a pornography store. I was buying pornography. Homer's Brain: Heh heh heh. I would'a never thought of that.
In the Life's A Zoo episode "It's Not Easy Being Chi", Ray explains what he is doing on the computer (he's actually selling Chi's belongings online) by claiming to be "downloading some internet porn".
In Iron Man: Armored Adventures, Tony often excuses himself from class (so he can become Iron Man) by claiming he needs to use the bathroom. Pepper later takes it upon herself to explain his absence to some crooks by explaining that he spends more time in the bathroom than anyone she knows.
In Adventure Time, Jake tries to get Finn out of an accidental engagement to the Slime Princess:
Jake: You shouldn't marry Finn. He pees his pants. Constantly. All the time.
Happens in one episode of SpongeBob SquarePants, where SpongeBob is forced to play stupid so that Patrick will look smart in front of his parents. The problem is, Patrick then starts to believe that SpongeBob really is stupid, and it all goes downhill from there.
In the Family Guy episode "Barely Legal", when Meg becomes obsessed with Brian:
Meg: I'm going to the mall later, maybe you can come and help me pick out some underwear. Brian: Uh, I don't think that's going to be a possibility, uh, I have plans, with Chris! Chris and I have plans this afternoon! Chris: We do? Brian: Yeah, yeah! We're doing that thing, we're doing what you usually do on a Thursday afternoon! Chris: Masturbate? Brian: That's it, that's what we're going to do together. Chris: Well, maybe back-to-back, but I gotta tell ya, I'm not a hundred percent on this.
Governor Mark Sanford of South Carolina went missing for a few days, claiming when he returned that he had been "Hiking the Appalachian Trail." Little did he realize (because that's not what he was really doing) that this just happened to be during National Nude Hiking Day. News outlets had a field day with that one, even before the real reason for his disappearance was revealed roughly a day later (a trip to Buenos Aries where he had been conducting an extramarital affair. Suddenly a new euphemism was born.)
This was a factor in the Lizzie Borden murder trial. She said that bloody cloths seen by a witness were menstrual cloths (they didn't have disposable feminine products back then). Everybody involved with the trial (all of them male) all got squicked out over this discussion of menstruation and decided to just accept that explanation and move on so they wouldn't have to hear about it anymore.