Question

What are the two formal duties of the vice president?

Answer 1

To cast a vote in the event of a senate deadlock, and to preside over and certify the official vote count of the US Electoral College

Answer 2

The vice president has two official duties.
One is to cast a tie-breaking vote in the event of a Senate tie vote. The Vice President is actually President of the United States Senate, but his only duty in the Senate is the tie-break vote.
The second duty is to officially oversee the vote count of the electoral college in the US Presidential Election.

Another important duty, of course, is to assume the presidency in the event that the President is rendered unable to continue his duties.