failed the nox in my emissions test.computer read a bad egr valve. after it was replaced the whole car was shaking and a bad burning smell came into the car so the mechanic reinstalled the old one .Could it be the cat or o2 sensor instead of the egr valve and if so why didn't the computer say so

yes its possible

NOX are oxides of nitrogen, the byproduct of burning hydrocarbons under high temperatures. This causes smog when it reacts with sunshine. The way that they overcame this was through the use of an EGR (Exhaust Gas Recirculation) Valve. This simply takes exaust and recirculates it back into the intake which cools the combustion temperature. This is the only device that they have to combat NOXs, the Cat does nothing against NOXs. It may have been possible that the EGR valve that the mechanic put on was no good and possibly stuck open (ours are electronic stepper motors, 3 valves, as opposed to vacume operated like most vehicles were). Two basic things you can do, lower your engines operating temperature and risk failing the hydrocarbons (by running a lower temp thermostat) or by trying another EGR. My first step would be to go with the EGR route.

I agree completely. The purpose of the cat is to knock down any unburnt hydrocarbons from a car running too rich, or with an ignition miss.

Your EGR is a prime suspect. It needs to be tested first. It may only need to be cleaned and re-installed.

Your mechanic may have put the wrong EGR in or one that was stuck open.