Q:

Why did Cobb ask Ariadne for the secret passage in the third-level dream?

In the third-level dream (the snow one), Saito and Fischer are compromised when they try to enter the building. Cobb asks Ariadne to tell him the secret entrance, but she refuses because if Cobb knows, then Mal knows. Cobb insists and finally Ariadne tells him, but she is the one that tells Saito and Fischer the secret entrance by radio.

Why did Cobb ask for the entrance?

Why doesn't he just ask Ariadne to tell the secret entrance only to Saito and Fischer, and not to him?

A:

There is no good reason. This is a known plothole. See here, among other places online. Nolan needed Cobb to know about the back-door in order for Mal to know about it, and was counting on people being too engrossed in the film to notice the flaw. One could argue that Cobb subconsciously wanted Mal to blow the mission, thus causing him to stupidly ask for the inside knowledge, but that is a weak argument.

A:

After rewatching the scene (again), it basically boiled down to running out of time.
The first level kick is already happening, they hadn't got Fischer in the room by the locked gate. Cobb forces Ariadne to tell the team the secret entrance (the air vent ducts) but in doing so (because Cobb was in earshot of Ariadne )  he obviously heard about the secret entrance too.
So telling just the team wasn't an option, maybe Cobb could have run over into the bushes and pulled his ear piece out or something but because they were panicking and rushing Cobb overruled Ariadne, forcing her to just tell everyone about the secret entrance.
Also another reason why Cobb asked and deemed it necessary is because of Cobb's character, which is even highlighted by his team mate Arthur in the film:

Arthur:  So now you've noticed how much time Cobb spends doing things he says never to do.

Cobb is so arrogant or reckless with his subconscious projections (Mal) that he thought it would be ok if he over heard the secret entrance.

