I understand how the Bitcoin network achieves its security through POW  mining i.e
making computers solve difficult mathematical equations, to achieve its reward while disregarding any bad players trying to cheat the system and thus making it a waste of time and money for them do attempt it.

but could someone please explain how POS and DPOS achieves this level of security? i haven't looked in to this yet, and don't quite understand it.

Also when the Anti-crypto brigade go on about how energy consuming crypto would be.... wouldn't it work out to be more energy efficient than printing, transporting and distributing all these ponzi schemed FIAT currencies around the world without mentioning the energy you would save not having to run these banks and even the petrol saved from employees not driving to the banks?