Dan is correct that you can petition the Board of Appeals at the County to revalue the home, however there is no guarantee. Ultimately, the County needs a certain tax base and make their valuation of homes based on the overall market and only take into account specifics of a particular property on a very case by case basis. At one point the Counties were trying to implement a tax valuation based on purchase price model because home prices were going up so fast they felt they were losing out on some revenues, however that was never implemented so they make their valuations based on an overall market view. Here's a link to the Pierce County website below this comment.
You can always petition the county to reassess the value of the home after the purchase, but no one can guarantee the outcome. It seems reasonable that a home assessed for a higher value than its most recent sale should be reassessed, but reason doesn't always achieve results.