Legally, Cruz doesn't have to prove anything. As he stands, according to law, he is a natural-born citizen and eligible to serve as President. Just as John McCain, George Romney and Barack Obama (so far as we know) were eligible.
For reasons of perception, he has renounced his claim on Canadian citizenship. But he didn't nee to do that -- it had no effect on his legal status.ib
What would you propose he needs to do...???
The Twentieth Amendment, Section Three places the burden of proving eligibility squarely on the President Elect. If he/she fails to qualify, Congress must name an interim President. That Congress failed to do it’s job twice with Obama did not mean that Obama became a legal President, only that Congress allowed a usurpation of the office of President. That they did not enforce the Twentieth Amendment, Section Three on Obama does not mean that they will not enforce it on Cruz.