Disabled monitor does not get disabled

I have a computer connected to a TV via HDMI. The TV has smaller resolution than the computer's monitor.

I want the screens be doubled when I use TV but not otherwise. All works well if I mechanically plug the TV: the resolution on the computer bercomes the same as on the TV and I can watch films on the TV. When I unplug, the resolution becames back to normal resolution of the monitor.

But I do not want each time plug and unplug the cable. Unfortunately Windows thinks the TV is plugged in even if it is powered off, and does not allow to set the normal resolution on the monitor.

If I disable the TV in the Device Manager, I get the opposite result! The computer's monitor becomes black and the picture is visible only on the TV! This is weird!

What should I do?

Alternatively I want to be able to set the resolution of the monitor to its native value manually even when the TV is plugged in!