A:Answer You can as long as there is a fuse to protect the curcuit.
Cutting the power cord may void the warranty though and/or
reverse the polarity destroying the unit and the warranty.
I spent a fair amount of time installing car stereos, CBs,
& etc, in my own business. I didn't want call backs or redos.
You may also be better installing an auxiliary "cigarette lighter" size outlet and use the normal power cord. I presume you want to hide the extra long cord? In this case the original cord plug can be hidden as well. That will give you the clean, hidden, uncluttered look as well. "Sleeper" high powered car stereos that use the stock speaker locations are highly advised to deter theft (as a hidden cord on the mirror).
No one will notice what you really have!
Good luck on your mirror project.
