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The service manual says to remove the dash, step 1 is to remove the steering wheel. I managed to remove the radio from an '89 by only losening the dash and not removing the steering wheel. It seemed to take years of fiddling with the dash after that to stop it from rattling. What I notice about old 89-92 Corollas is that the mechanical parts are often in excellent condition, but the body parts are often in terrible shape because former owners have tried to do their own anti-theft, window repair or audio work. So, if your car is somewhat beat-up on the inside then I suppose it doesn't hurt to replace the radio in a "field expedient" fashion. But if it is nice, better do it by the book.
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