John Crichton said:
You do recall (as I'm assuming you do) that it was a setup, right? He wasn't trying to keep it a secret. He wanted to frame Borden, so no witnesses could be there.
Actually, while that was my first thought, by the time I got to the end of the movie, I no longer believe that to be the case. I don't think he was concerned about framing Borden- all he cared about was being able to outshine him and create a better magic trick. The fact that he blew Borden's mind and curiosity, and that he ended up being framed for the crime was, IMO, incidental.
As to what makes me think this- well, recall that he did go to find Tesla to have him make him a trick like Borden's. While that was just a red herring planted by Borden, he at least seemed to think that Borden was doing the same thing (or something similar). His version of it was more dramatic, though, because he was the greater showman.
I do think he was probably tickled that things ended up the way they did- with Borden being framed- and the reason he tore up Borden's confession about the trick was because he didn't really care why he did it- he had won. By having Borden reveal his ignorance of the trick (by sneaking backstage as he did), he knew that he had a secret Borden did not. The only reason he wanted Borden's "secret" was to show that he had proven himself the better. By forcing his confession, he was the greater magician, and he didn't need to know the secret.
That being said, I do think that he was taking Borden's daughter not out of spite and vengeance, but because it wasn't as if he could just reveal himself to be alive (and all the questions that would lead to- including the outing of his secret trick), but because circumstances unfortunately would have led to her being an orphan (he didn't know about the brother) and his sense of compassion wanted her taken care of.
Well, I'm sure there was a little bit of vengeance involved, but Angiers- twisted as he became- was still not wholly a bad person, and I think he would want to make sure that Borden's daughter didn't suffer any more due to the pettiness between the two men.
*shrugs* I don't think it really matters to the story but Angier did have many resources. He could have had other people around that made sure that only Borden was allowed back there and no one else. And he was already committing murder/suicide every night anyway so they guy is already a but shaky in the brains department.
He did have some sighted people standing guard- Borden has to get past one of them in order to get backstage. He takes of his fake beard to "prove" he's part of the show- something you wouldn't do for a blind man.
It's just that Angier evidently didn't hire anyone terribly competent. (Although, they probably wouldn't have let Cutter backstage, as they would have recognized him.)