2005 trailblazer headlights go off ocasionally

the click you are talking about is the high beam relay.
you nedd to get a tester or voltmeter and check the sockets for 12v.
there has been instances where both low beam bulbs went at once.
do the daytime running lights work?
if not,supect the daytime module...it is also wired into the parking brake.
if you put the park brake on first,then start the vehicle,the dtr do not come on