I am a student in contact with a business owner, and I am having a hard time deciding on a complimentary close. Should I use it for every email I send him? I don't want to make it redundant since it is for formality's sake.
For formal emails/letters to a business owner/principal/someone of authority, you'd use "Yours sincerely" or "Yours faithfully", followed by your name on the next line.
"Sincerely" is used when you don't know the person's name. (ie. Dear Sir; Yours sincerely)
"Faithfully" is used when you know the person's name. (ie. Dear Mr Smith; Yours faithfully)
To make it easier to remember, you can just remember than the two "S" don't go together. (S from Sir and S from Sincerely)
If I were you, I'd just go with "Dear Mr xxx" and "Yours faithfully" every time I send an email. It won't be "redundant" as you call it. Many things in emails and everyday life are redundant. However, we still use it. Also, it's not redundant if it shows him how polite you are. It's better to be too formal than too casual. Being formal would also show him how serious and sincere you are in contacting him.
Hope this helps.