If you mean the name Giorgio Armani, this is custom drawn.

In college I used to work at Armani Exchange, and I remember seeing a style guide saying that the Armani logo font was Bauer Bodoni manually condensed. I don't remember the actual percentage condensed. But that damn spur on the G must have been customized. Also, I saw this style guide back in 1993, and I believe the switched to a Didot face a few years ago, especially since Fabien Baron (ex Harpers Bazaar art director and Didot fan) does a lot of graphics work for them. My personal belief is that they now use a customized version of Didot for the logo and HTF Didot for their sub-brand names, like "armani mania" perfume, which you can tell by looking at the lowercase a.

If you just want something similar, you could use almost any Bodoni and add a spur to the G and be very close.

another suggestion might be to use Didot from Hoefler & Frere-Jones. Didot-HTF-M06-Medium to be closer.

I am actually trying to find the font used for “cosmetics”. Is that the same as “ARMANI”..??

that depends on the top of the lowercase t. is it scooped, angled or flat?

Looks more like Didot than Bodoni — the G already has the spur and the shapes of the s and R in the sample are closer to Didot.

Specifically, have a look at Italian Didot