how does a double insulated appliance trigger a rcd don,t it need a earth wire to trigger it?

Answer 1

Actully you don't need an earth wire because the RCD operate when it sense a different in current
for example you have a double insulated appliance which protected by RCD the RCD makes sure that the current going to appliance is the same as the one returned from the appliance if there is any leakage current for any reason the RCD tripped
You must know that you must protect unearthed installation with RCD devices

Answer 2

The principle the RCD works is that it is monitoring leakage currents from a live conductor to earth. So yes you definitely need and earth wire to trigger it.
Not RCD cannot be used where your Neutral and Earth conductors form a common wire. It will be tripping at all times.