How do you replace drivers airbag on 2002 mercury cougar?
You don't. This is not a DIY repair. Working on the SRS can be very hazardous and is best left to a professional.
It is an open circuit in the drivers airbag. There is a clockspring behind the steering wheel that may be worn. To replace, remove airbag, steering wheel, column covers, and lower dash cover. MAKE SURE WHEELS ARE STRAIGHT. The new clockspring comes centered with a yellow locking tab. Remove that after installing it.
I assume that you mean the drivers side, steering wheel airbag. First, remove the battery from the vehicle and leave it for 30 minutes. There should be four torx bolts on the back of the wheel, these are recessed pretty deep, so you'll need to get a long driver to remove them. After that, it should come off just fine. Remember, airbags are dangerous, remove them at your own risk.