What to do if you’ve put the wrong last name?

I have made a terrible mistake, both my parents are separated and because of that I have been known by myself and other by both their last names hyphenated. However, I didn’t find out until after I applied for student finance that my last name legally was my fathers.
I have to prove to student finance that I am not an imposter for giving my legal documents (which I have only just seen) that have my legal name and not my known name.
A side note is that my family hasn’t got the time or money to legally change my name at the moment so I still have that name.
What is an easy, cheap way to prove my identity as my known name and my legal name?