Disconnect the power
Start by turning off the breaker, and pulling the serviceman disconnect, which will typically look something like this.
This will insure no electricity is flowing to the condenser unit while you're working.
Open the unit
Next you'll want to disassemble the unit, to allow access to the electrical parts. This will vary from unit to unit, so check the owners manual for the procedure for your unit. Once you have the unit opened up, make sure to discharge the capacitors.
These things store enough power to kill you, so you don't want them to discharge accidentally.
Resistance is not futile
Once the power is completely removed from the unit, it's safe to start poking around (electrically speaking, don't go busting the refrigerant lines). Start by tracing the wires from the condenser fan motor, back to where they connect in the electrical box. There should be 3 or 4 wires. In my unit, I had Black, White, Brown, and Brown with a White stripe (your model may vary). To determine if the motor is good, you'll measure the resistance across each coil. To do this, you'll have to disconnect the wires, so the motor is no longer part of the circuit (make note of where the wires connected).
Typically you'll have 3 wires, start, run, and common (we'll ignore my 4th wire in this answer). Set your multimeter to measure Ohms, and start measuring. You're going to measure the resistance between each combination of two wires to determine what each wire is, and if the motor is still good. Let's start with Black and White...
Black -> White = 15.9
Black -> Brown = 35.4
Brown -> White = 51.2
Knowing that...
Common -> Run = Lowest resistance
Common -> Start = Medium resistance
Start -> Run = Highest resistance
We can determine that...
Black = Common
White = Run
Brown = Start
If we also know that the two lower readings should always add up to the larger reading, we can safely say this motor is still good. If you measure 0 or infinity between any pair, that means you have a shorted or an open winding and the motor should be replaced.
Repeat the same procedure for the compressor motor.
Shorts on the ground
The other thing you'll want to check for, is shorts to ground. Set your multimeter up to test impedance. Put one probe on the equipment grounding conductor of the feeder, and the use the other to find a solid ground on the motor. You may have to scratch some of the paint off, especially on the compressor. Once you've found a solid ground, measure from each motor wire to your ground spot. If the meter beeps or give a low resistance reading, you have a short to ground. As with the resistance test above, the motor should be isolated from the circuit when doing this test (once a solid ground is located).
Does it really need (not should) to be replaced: Is there a lifetime failure count for a breaker? -StackEx
Here's some extra Yahoo answers nonsense, from my attempted search in finding a way to safely make a breaker pop.
The other answers to your question here bring valid concern about replacing a questionable breaker. I'd be more concerned with the condition of the wire, which I believe was the noise (rattling in the pipe). Circuit breakers are pretty good about failing in a safe state, I.e, it just won't work anymore. I wouldn't be so concerned that a monetary/slight (crappy) contact didn't trip the breaker; you did not achieve a dead enough short (see wallyk's). It was not a good thing to have happened (although not the worst) and if you're not going to pull the wire it may be safer to replace the breaker with an arc fault.
THQL1120AF, an arc fault for $50, except it's not a tandem. No good unless it happens to be the one single you have there. aplussupply.com sells standard replacement breakers for your panel. I was unable to find a THQL1120 (AF or GFCI) tandem breaker. FYI, I looked up a 100amp AF for that panel, it's $800...
You could swap that single for a new regular tandem, freeing an entire slot (by taking the other wire from the bad breaker (provided that it was on the same leg originally) for a 1" AF, now hooked to the circuits' questionable wire. Request clarification if you do not understand the importance on which of the legs a breaker is on and how to properly relocate them.
Go big or go home. Meaning, as discussed here, replace that panel or leave it alone as much as possible.
Best Answer
Breakers are supposed to be good for a fairly large number of triggers, and they are supposed to fail open when they fail. Is there a chance that you were running at a heavier overall load when the second incident occurred? I'm thinking that if you were say within 10 amps of max on the main breaker just due to normal load (air conditioners, perhaps?), and caused a short on the secondary (a 15 amp circuit, perhaps), you might have triggered the main to trip first by simply overloading it, before the secondary could draw enough to trip.
If you really think the secondary might be bad, then by all means replace it, but no, you don't have to replace the main just because it tripped. As long as it feels normal when you move it back into position (flip some other breakers to see how they feel - there shouldn't be any slop in the movement), you'll be fine.
The last several times I've seen breakers fail they've all failed open.