How do you know if your ac compresser is bad on a 2005 trailblazer?

If a compressor goes out, it tends to be pretty obvious. Something might seize, there might be smoke, there might be a loud squealing sound, etc. IIRC, the TrailBlazer uses a cycling clutch compressor, so it should cycle off when the pressure reaches a certain point, and cycle back on when it gets reduced to a certain point. If it fails to do that, try disconnecting the low pressure switch and jumping the wires. If that doesn't engage it, then you probably have a bad low pressure switch. If that doesn't solve the problem, try hooking a manifold gauge set to the AC system... if your pressure readings are particularly low, especially on the high pressure side of the system, then a defective compressor is a possibility, although there are other possible causes, as well. Narrowing it down requires a pretty comprehensive knowledge of AC systems, and you're probably better off to let a qualified automotive technician handle this.