Hi DAVEEB
Yes this problem has become prevalent in the last 20 years (Mind you if you have ever watched a 405 LINE TV you'll know what a 10 khz whistle is like
).
The LINE Scanning circuitry on TV'S and MONITORS has always been prone to this , but the problem has been exascerbated by the use of high frequency power supplies (SMPS) in TV'S MONITORS and indeed computer power supplies .
The whining is usually caused by RESONANCE of iron cored inductors (TRANSFORMERS ETC )at particular frequencies in the LINE stage or POWER SUPPLY stage within the monitor.
When the monitor was manufactured this resonance is prevented by using a resinous material on the transformer cores to damp the oscillations. Howver as the monitor ages this resin dries up and can't damp the oscillatiobs any more.
The problem can be TEMPORARILY reduced by using different RESOLUTION settings ( As ADEPT has suggested ). However you will find that this only temporarily solves the problem
The common "FIX" is to scrape off the old defunct resin material from the offending inductor and apply a layer of new material ( ARALDITE being ideal )
So really , unless you are confident enough with these things to "HAVE A GO YOURSELF" then you either have to accept it or get an "EXPERT" to do it . ( EXPENSIVE
).
Hope this is of some assistance .
Cheers
Scrumpy-Jack