The documentary, How to Crack the Class Ceiling, was broadcast in December last year and, around 11 minutes in, features Rajan delivering a piece to camera from the back of a cab.
The presenter can be clearly seen not wearing a seatbelt, which is against the law in the UK when the safety device is available.
The BBC said: “In the course of making this programme the presenter was filmed travelling by cab without using a seatbelt. This was a mistake for which we apologise.”
Rajan appears to be traveling through London in the clip. The Metropolitan Police website states that people can be fined up to £500 ($606) for not wearing a seatbelt.
The police force has been contacted for comment. Rajan has also been contacted for comment.
The BBC apology comes two months after British Prime Minister Rishi Sunak was fined by police for not wearing a seatbelt. It follows Sunak’s team posting a social media video featuring the PM not wearing his safety device in Lancashire. Downing Street said it was a “brief error of judgment.”
Rajan is one of the BBC’s highest-profile presenters. He presents Radio 4’s Today show, the flagship breakfast news program, and is the new host of University Challenge, one of the longest-running quiz shows on British television. He is paid up to £330,000.