When you cosign a document, you sign it with another person. You and your business partner might cosign a lease for your new office space.
You can use the verb cosign in two slightly different ways. First, if two (or more) people put their signatures on a form or letter, you can say they cosign it. In some cases — particularly when you’re applying for a loan — a financial institution will require that you have someone cosign, essentially promising that if you can’t repay the loan, he or she will be responsible for it. The word dates from the mid-20th century.