Yeah, they can't do that sort of thing - reminds me of a story told by a mate who used to be in the army (and used to have first a mental twin-turbo Renault 21, then a tricked out Golf GTi... before the divorce) of a similar thing that happened to him.
Quite a while ago... Driving normally in the Renault, going home on leave from the army base, dark winters evening(?), not rushing at all (as he knows the car is a race and cop magnet), someone starts tailing him... he gets a bit paranoid (its around the time of the worst IRA campaigns) and progressively ups the speed to see if they're just normal folks following too close, and fall off much past the limit. Or even cops looking for someone to book - he can probably talk his way out of it if the lights come on.
When they stay with him through some several random turns, up to 90 on the straights, and through a red, he floors it, bricking it. Then about 2 miles later, after they've been well over the ton on single carriageway roads and through a couple more lights.. the blues come on.
He's relieved by this, but also very pissed off at their behaviour and knows his rights. Belt off, and brakes hard into the next layby, quick handbrake turn to point the car back at them (shining headlights on the cars occupants, and cover if needed behind door), jumps out while engine still running and smoke rising and waits for them to stop, oldskool car-phone in hand, dialling the non 999 number for the local station.
You two! I want your names, badge numbers, and the name of your supervising DC. NOW.
Turned out they saw what they knew to be a powerful car, and were expecting it to be driven hard - easy money. He didn't immediately oblige but for some reason they stayed with him, not counting on the somewhat obvious point of anyone coming off a military premises being understandably nervous of anyone popping up from nowhere and following them close. His nervousness carried through into an adrenaline trip of getting someone high-up on the phone, himself later absolved of any charge, and the two officers in some sticky schtuck.
Just too bad there weren't any GATSOs or many red light cams about back then. Getting proof of that chase on camera would have been sweet.
Why have they done wrong (in both cases)??
1. Yep, entrapment. If they weren't racing you and goading you on, chances are you might have driven perfectly legally. The officers have pushed you into a crime.
2. Wilful dangerous driving and unneccessary pursuit... its just lucky it was a quiet road, and quiet junctions.
3. They too were committing an offence. Anything a police traffic officer does in their car without the lights and/or siren on counts as civilian action. They're not allowed to speed, run reds, etc, unless those beacons warning other road-users and marking it as an emergency vehicle are switched on. Even more so for an unmarked car. So if they have that video, and it doesnt have that little icon in the corner showing that the blue lights are on.... but it does show the speeding and the light-running aspects of the race... ask em for a copy of it, their numbers and what OCU they're based at. (you might not get any of that, but possibly some ashen faces if they realise why

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In both your case and that of me mate they should have turned the blues on as soon as it was clear an offense was being committed, i.e. when he passed 75 (enough for points on a 60-road), or at least after the first red light was run - but they kept going to higher speeds through more potential certain-death intersections. Dereliction of duty and all that. Even if you didn't stop (which i'm sure both would have) it would have warned others that something's afoot, lowered the risk of an accident, and covered their own asses legally.
At least your lot were pretty cool about it - just some bored guys out for a blast and something amusing to take home!
(you dont work at a night club do ya?

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Calvin, man...... you're a nutter.
Keep out of brum unless you're in a 1.0 now, yhear?
