Lage counting on 'hard' man Costa to give Wolves some bite in the box
Wolves signed former Chelsea striker Costa on a free transfer after they lost both Sasa Kalajdzic and Raul Jimenez to injuries. New recruit Kalajdzic is expected to be sidelined for nine months after knee surgery.
Lage's side have scored only three Premier League goals this season - the lowest in the top flight - ahead of Saturday's home game against champions Manchester City.
When asked if his side had been too nice in the past and whether Costa would change that, the Portuguese manager told reporters: "That's a good point. If you want to press and be more aggressive, sometimes you cannot be too nice.
"You need to be aggressive in the duels and he (Costa) said that to me. 'I know you ask about my profile, my personality, I am one of the best guys outside the pitch but inside the pitch I am very hard'.
"That is the kind of man I want also. It was a simple talk and it was very easy to convince him to join us."
Costa has not played since leaving Brazilian side Atletico Mineiro in January, however, and he admitted he would need "two to three weeks to get back in shape".
"We are very happy with him. We know he needs time not to be fit, because he is fit, but to be ready to compete at this level. It is his destiny (to return to the Premier League)," Lage added.
"Fifteen months ago when I think about him we didn't have any chance (of signing him), now he said to me what happened is because it is meant to be.
"We need a reference in the box, we need one guy to compete with Raul and Diego has that profile, and he has a different profile to Raul."