ILANA DIAMOND ROVNER, Circuit Judge.

Thomas Jensen conditionally pled guilty to one count of mail fraud. He reseiwed the right to appeal the district court’s refusal to suppress evidence seized from a car in a warrantless search, and his resulting confession (the fruit, he claims, of a poisonous tree). We affirm.

Thomas Jensen was returning merchandise to a Best Buy store in North Riverside, Illinois, when he caught the attention of a *1045 store security guard. The guard recalled that Jensen had been in the store a few days earlier returning merchandise, and at that time had printed one name on a return receipt and then signed another name. The guard suspected fraud, and called local police. Sergeant Anthony Garvey and Officer Debra Ecklund responded to the call. The officers and the security guard approached Jensen and asked if they could speak to him in the store security office. Jensen agreed, but as the group approached the office, Jensen bolted for the exit. The officers pursued him, chasing him first into a nearby Wendy’s restaurant, where he pushed Officer Ecklund as he ran out again. He next charged into Sally’s Beauty Supply, and unwittingly trapped himself in the employee lounge at the back of the store, from which there was no exit. Sergeant Garvey arrested Jensen on charges of battery and criminal trespass to property, and other officers transported him back to the police station.

At the station, Officer Ecklund read Jensen his Miranda rights, and he signed a waiver of those rights. He explained to Officer Ecklund that he was returning an item he had purchased at a Best Buy in Colorado. He told her that he had driven from California to Illinois in a car owned by his stepfather, Donald Matthews. He claimed to have run from the police because he was seared.

Officer Ecklund contacted Matthews at his home in California later that evening, and confirmed that he owned the ear. She also verified that Jensen had Matthews’ permission to drive the ear. Matthews asked Officer Ecklund where his car was, and she informed him it was in the Best Buy parking lot. He asked what would happen to the car, and Ecklund told him it was up to him. He asked if the car could be towed and what the cost would be, and Ecklund told him the price of a private tow and daily storage. Matthews then asked if the police could somehow secure his ear until he could send his other son to Illinois to pick it up. Officer Ecklund responded that, with permission from her sergeant, she could drive the car to the police station and store it there. She also told Matthews that if she drove the ear to the station, the police would have to inventory the car in order to protect themselves. Matthews replied that the inventory “wasn’t a problem,” and “that would be great.”

In the meantime, after Sergeant Garvey learned about the car from Officer Ecklund’s interview with Jensen, he returned to the Best Buy parking lot where he found a 1993 Volvo matching Jensen’s description of the car. Because the officers had initially been called in to investigate a possible fraudulent return, Sergeant Garvey decided he would like to search the ear. 'He returned to the station to question Jensen about the car at approximately the same time that Officer Ecklund was talking to Matthews. Unaware of Eeklund’s conversation with Matthews, Sergeant Garvey approached Jensen and asked for his consent to search the car. Jensen replied, “Well, I can’t give you permission to search that car because it’s my stepfather’s car.” Garvey ended the discussion at that point.

Later that evening, Officer Ecklund told Sergeant Garvey that she had spoken to the owner of the Volvo, who had requested that the police drive the car to the station and secure it until he could have another son fly out to pick it up. Garvey asked if Matthews had been told the car would be inventoried under those circumstances, and Ecklund confirmed that Matthews understood this and agreed to the procedure. Sergeant Garvey then picked up the car and drove it to the police station. Over the course of the next day, police officers took an inventory of the extensive contents of the car, including thousands of dollars worth of software, credit slips from Best Buy, detailed logs with credit receipts, approximately $8000 in cash and travelers’ checks, and approximately $4000 in Office Depot cheeks.

The North Riverside police alerted the FBI, and two special agents arrived the next morning to interview Jensen. When confronted with the mound of evidence seized from the car, Jensen confessed that he had been engaged in a two year scheme to defraud retail stores across the country. The scheme involved purchasing merchandise at reduced prices, which Jensen accomplished primarily by switching price tags. He would then return the merchandise for a full refund. The stores often refused to' issue cash *1046 refunds, instead insisting on mailing refund checks to Jensen. Jensen used a number of different names and addresses to receive the refund checks. Because Jensen used the mails to accomplish his scheme, federal authorities eventually charged him with mail fraud. Jensen was arrested on federal charges approximately one month after he was released on bond from the local charges of trespass and battery.

In the district court, Jensen moved to suppress the evidence seized from the Volvo and his resulting confession. He contended that the officers acted without a warrant, and without his consent. After holding a hearing at which Jensen, Matthews, Ecklund and Garvey all testified, the district court denied the motion, and upheld the search on four different grounds. United States v. Jensen, 1997 WL 610462 (N.D.Ill. Sept. 19, 1997). First, the court found that Jensen abandoned his interest in the car- and had no reasonable expectation of privacy in its contents. Second, the court found that the search was conducted pursuant to consent from Matthews, the owner of the car. Third, the court ruled that the search was valid under the inventory exception, in light of Matthews’ request to the officers to take the car under their care. Fourth, the search was allowable, according to the district court, because the police could have seized the car under their community earetaking function, at which time the inventory exception would again apply. On the basis of this ruling, Jensen pled guilty conditionally, reserving his right to challenge on appeal whether the search of the Volvo was conducted constitutionally.

II.

Jensen raises five separate, issues on appeal. First, he faults the district court for crediting the testimony of Sergeant Garvey and Officer Ecklund, testimony that he characterizes as “exceedingly improbable.” Second, he contends that the court erred in concluding that he had abandoned the car and thus lacked standing to bring a Fourth Anendment claim. Third, he asserts that the court erred in finding that Matthews’ consent to search the car was valid. Fourth, he contests the district court’s ruling that the police department could have legitimately seized and searched the car under its community caretaking function. Finally, he posits that if the search of the car was improper, then his confession should also have been suppressed under the “fruit of the poisonous tree” doctrine. Of course, if we affirm any one of the district court’s rationales for upholding the search, Jensen’s appeal fails in whole.

We consider Jensen’s challenge to the court’s credibility determinations first, for much of the district court’s reasoning rested on the testimony of Sergeant Garvey and Officer Ecklund. We review the district court’s credibility determinations for clear error. Fed. R. Civ. Pro. 52(a); Anderson v. City of Bessemer City, N.C., 470 U.S. 564, 573, 105 S.Ct. 1504, 84 L.Ed.2d 518 (1985). 1 We defer to the trial court’s credibility determinations because “only the trial judge can be aware of the variations in demeanor and tone of voice that bear so heavily on the listener’s understanding of and belief in what is said.” Anderson, 470 U.S. at 575, 105 S.Ct. 1504. When a trial judge credits the testimony of one witness over another, “each of whom has told a coherent and facially plausible story that is not contradicted by extrinsic evidence, that [credibility] finding, if not internally inconsistent, can virtually never be clear error.” Id. See also, United States v. Stribling, 94 F.3d 321, 323 (7th Cir.1996) (because the resolution of a motion to suppress is fact-specific, appellate court specially defers to the court that heard the testimony and observed witnesses at suppression hearing); United States v. Veras, 51 F.3d 1365, 1370-71 (7th Cir.1995), cert. denied, 516 U.S. 999, 116 S.Ct. 540, 133 L.Ed.2d 444 (1995) (no clear error where district court finds testimony of one witness more credible than another, and the finding is supported in the record); United States v. Cardona-Rivera, 904 F.2d 1149, 1152 (7th *1047 Cir.1990) (trial judge’s choice of who to believe is binding on the appellate court unless the judge credits exceedingly improbable testimony).

Jensen maintains that the testimony of Officer Ecklund and Sergeant Garvey was exceedingly improbable in three respects. First, Jensen complains that Garvey’s story is fatally flawed (indeed, Jensen calls Garvey’s testimony “insulting”) because no veteran police officer who wanted to search a car would simply give up when a suspect denied he had authority to consent to a search. According to Jensen, Garvey would have given up that quickly only if Jensen had unequivocally refused the search. Of course, the district court found that Jensen did not flatly refuse the search but rather stated to Garvey, “I can’t give you permission to search that car because it’s my stepfather’s car.”

Jensen’s second challenge to the officers’ credibility arises from an alleged inconsistency between the testimony of Garvey and Ecklund. According to Jensen, Garvey testified that he spoke to Ecklund before he sought Jensen’s consent to search the car, and that he spoke to her again before she wrote her police report. Jensen posits that Ecklund contradicted this testimony by stating she did not talk to Garvey before he spoke with Jensen, and that she was unaware he was going to speak to Jensen. The district court did not address this alleged inconsistency. Our review of the record reveals that Garvey and Ecklund had different recollections of the timing of their conversations during the evening of Jensen’s arrest. However, both recounted the essential contents of their conversations, and their testimony about the substance of their discussions was consistent.

Third, Jensen attacks Officer Eek-lund’s credibility because her written report did not reflect Matthews’ consent to the inventory search of the Volvo. Rather, her report simply noted that she “advised Mr. Matthews that we would go locate his vehicle, drive it back to NRPD, inventory all the property inside and secure it, until he could make arrangements to have it picked up.” Jensen further contends that the destruction of a tape recording of Ecklund’s call with Matthews casts further suspicion on Eek-lund’s testimony.

The question before us then is whether any of these “discrepancies” render the testimony of Sergeant Garvey and Officer Eek-lund “exceedingly improbable.” Under the Supreme Court’s formulation in Anderson, we must determine whether the testimony was coherent and plausible, not contradicted by extrinsic evidence, and not internally inconsistent, because a district court’s finding that such testimony is credible can virtually never be clear error. Anderson, 470 U.S. at 575, 105 S.Ct. 1504. We are hard pressed to find any internal inconsistency or implausibility in Sergeant Garvey’s testimony about his attempt to obtain consent to search from Jensen. He merely testified that he asked the suspect for consent to search his car, and that when the suspect told him the car was not his, he ceased his efforts. There is nothing implausible about this story. We are often faced with appeals from defendants who claim that police officers forced them to consent to searches of their homes or vehicles with improper threats or physical intimidation. But we have to admit that this is the first time we have heard a defendant complain that the police did not badger him enough. To contend that a police officer’s testimony is not credible because the officer behaved lawfully is a frivolous argument. The district court did not clearly err in crediting ■ Sergeant Garvey’s testimony on this point.

Nor do we believe that the inconsistency in the testimony of Sergeant Garvey and Officer Ecklund regarding the timing of their conversations renders the remainder of their testimony exceedingly improbable. This minor inconsistency was but one factor that the district court considered in assessing the credibility of the officers. The district court also observed the variations in demeanor and tone of voice of all of the ^witnesses to testify at the suppression hearing, and obviously discounted this minor difference on an insignificant matter. See Anderson, 470 U.S. at 575, 105 S.Ct. 1504. Witnesses are not incredible as a matter of law simply because they have been impeached on trivial, irrelevant matters. United States v. McEntire, *1048 153 F.3d 424, 435 (7th Cir.1998) (discrepancies arising from impeachment do not render witness testimony legally incredible). Crediting the remainder of their testimony was not error at all, much less clear error.

Finally, we find nothing inconsistent or implausible in Officer Ecklund’s testimony about her conversation with Matthews. Her failure to note Matthews’ consent to the inventory search is unremarkable. As we discuss below, Matthews’ consent to the inventory search was not necessary, and Ecklund could well have considered his consent irrelevant and not worth noting. Nor are we moved by the erasure of the tape recording of Ecklund’s conversation with Matthews. Jensen does not deny that the North Riverside police department routinely held such tapes for only thirty days before reusing them. The North Riverside police had no reason to keep this particular tape because Matthews’ consent was irrelevant to the state charges of trespass and battery. Only the federal authorities had a reason to preserve the tape, and they did not bring charges against Jensen until more than a month later. By the time the FBI requested the tape, it had already been destroyed. None of this renders Ecklund’s testimony exceedingly improbable, and we therefore reject Jensen’s challenges to the district court’s credibility determinations.

Accepting the facts as the district court found them, Jensen did not flatly refuse to consent to a search of the Volvo, but rather told Sergeant Garvey that the car belonged to his stepfather, and that he could not, therefore, consent. Moreover, Matthews asked Officer Ecklund to take custody of his car, and when she told him the police would conduct an inventory search of the car, he consented to that search. Given these facts, we now address whether the warrantless search of the Volvo fell within the bounds of the Fourth Amendment. Jensen argues that it does not because he did not abandon his privacy interest in the Volvo. He contends that even if we find that Matthews consented to the inventory search, Jensen’s privacy interest in the car was superior to his stepfather’s interest, and thus Matthews’ consent was not valid against Jensen’s assertion of his rights. Jensen implicitly argues that his superior privacy interest in the car was also enough to overcome the inventory exception to the Fourth Amendment.

Consent to a search is a well-recognized exception to the need for a warrant or even probable cause, and we know that Matthews consented to the search. See Schneckloth v. Bustamante, 412 U.S. 218, 219, 93 S.Ct. 2041, 36 L.Ed.2d 854 (1973). Warrant-less inventory searches of cars in police custody are also proper as long as the police lawfully have custody of the vehicles. United States v. Sholola, 124 F.3d 803, 818 (7th Cir.1997). 2 We know that the police lawfully had custody of the car because the owner here asked the police to take custody. The sole issue then is whether Jensen had some privacy interest in the car that exceeded Matthews’ to such an extent that the police search was unreasonable under the Fourth Amendment. We conclude that he had no such superior interest.

Jensen’s statement that he lacked the power to consent to the search because the car belonged to his stepfather indicated to the officers that, in the very least, he and his stepfather had common authority over the car. See United States v. Matlock, 415 U.S. 164, 171, 94 S.Ct. 988, 39 L.Ed.2d 242 (1974). In Matlock, the Court held that “when the prosecution seeks to justify a warrantless search by proof of voluntary consent, it is not limited to proof that consent was given by the defendant, but may show that permission to search was obtained from a third party who possessed common authority over or other sufficient relationship to the premises or effects sought to be inspected.” Id. The Court explained that the authority which justifies the third party consent rests on “mutual use of the property by persons generally having joint access or control for most purposes, so that it is reasonable to recognize *1049 that any of the co-inhabitants has the right to permit the inspection in his own right and that the others have assumed the risk that one of them number might permit the common area to be searched.” Id., 415 U.S. at 171 n. 7, 94 S.Ct. 988. See also, United States v. Duran, 957 F.2d 499, 503-04 (7th Cir.1992); United States v. Chaidez, 919 F.2d 1193, 1202 (7th Cir.1990), cert. denied, 502 U.S. 872, 112 S.Ct. 209, 116 L.Ed.2d 167 (1991).

We explained in Chaidez that the foundation of third party consent is assumption of risk. 919 F.2d at 1202. For example, a person who shares a ear with another person understands that the partner may invite strangers into it, and that the privacy right of one is not absolute but contingent in large measure on the decisions of the other. Id. “Decisions of either person define the extent of the privacy involved, a principle that does not depend on whether the stranger welcomed” into the car turns out to be a police officer or another party to the crime. Id. (emphasis in original). In this case, Jensen told the police that he shared the car with his stepfather, who was the actual owner. Jensen expressly acknowledged to the officers that Matthews had the right and power to consent to a search of the car. As the district court found, Matthews consented to both a seizure and a search of the car, and in fact affirmatively asked the officers to assist him by taking custody of his car.

Jensen complains that Matthews was two thousand miles away at the time, and thus the officers could not have reasonably believed he held equal privacy rights in the car. But Jensen essentially told the officers that his stepfather’s privacy interest in the car surpassed his own when he explained that he had no power to consent'to a search of his stepfather’s car. He cannot now be heard to complain that the officers took his word for it. Indeed, the officers did more than take his word for it; they confirmed with Matthews his ownership and equal control over the car. Having heard these statements from Jensen and Matthews, the officers reasonably believed that Matthews had authority over the car. See Illinois v. Rodriguez, 497 U.S. 177, 185-86, 110 S.Ct. 2793, 111 L.Ed.2d 148 (1990). The officers’ reasonable belief that the person consenting to the search had authority to do so is all that is necessary for a consent search to be valid, and the district court therefore properly denied Jensen’s motion to suppress. Id. Because the search of the car was proper, the district court was also correct in refusing to suppress Jensen’s subsequent confession.

Affirmed.

. Although Anderson was a civil case, "the same standards apply to the appellate review of fact-findings made in deciding a motion to suppress evidence in a criminal case.” United States v. Cardona-Rivera, 904 F.2d 1149, 1152 (7th Cir.1990).

. Consent to search is not necessary for an inventory search; all that is necessary is that the police lawfully had custody of the vehicle, and were acting pursuant to an established police policy. United States v. Lomeli, 76 F.3d 146, 148 (7th Cir.1996). That the inventory search may also have had an investigatory purpose does not invalidate it.. Id.