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Bradley Cooper and girlfriend Zoe Saldana have been living together for months, and here are the pix to prove it!

We've seen the twosome at Cooper's LA home many times ... morning, noon and night.

An X17 photographer tells us:

    "These two really don't want to go public with their relationship for some reason. They never ride in the same car, they're never seen together in public. They're such a great-looking pair, I don't know why they hide."

Until now, Brad and Zoe have been very discreet ... making sure to not leave or enter the house together.

So much for keeping their living arrangements on the down low.