President Obama didn't have to do that, and he certainly did not neglect taking care of his citizenship status. President Obama was born in Hawaii in 1961, AFTER it became a state (1959).
Cruz-the-fuse was born in Canada, which, in spite of the xenophobia of the radical right, IS a foreign country. That they don't care in his case reflects that their real objection was clearly racist bigotry.
Radical Right = HYPOCRISY (also bigotry and intolerance, and frequently racism). It would not surprise me to see some on the right objecting to Cruz because he is half-Cuban, and therefore unacceptable to them as an Hispanic/Latino (not 'white' enough for them), in spite of the fact that there are white and non-white Hispanics.
The irony is that no one is more of an intolerant ass than Cruz himself.