2002 Toyota Corolla white smoke.

During cold weather it is normal for some white smoke to exit exhaust, it would dissipate quickly. If it lingers and does not dissipate quickly, it is coolant.

This would indicate a head gasket, intake manifold leak. Have a compresion test performed.

If the car has recently, or ever, overheated, that can cause a head gasket failure.

Valve seals would leak oil, and cause blue smoke.