Matt
Tachographs are a massive grey area.
If you tow anything commercially within 50kms of your trading or home address then no you do not need a tachograph.
If you are anywhere over 50 kms then yes you need a tachograph - but it is dependant on what you are pulling the trailer with.
With a car you have more chance of getting away with it but I know from personal experience and because VOSA told me directly that they target van trailer combinations because they know they generally do not have tachographs fitted!!
I can't answer the technical part of the question as I work on an attitude of 'if its broken it goes in for repair' and I don't need to know the finer details.
Cheers
Rob