No, The Taken King is not part of the Digital Upgrade Program. So if you buy it for Xbox 360 you will have to re-purchase the expansion for Xbox One.
Also the expansions are not backwards compatible, which means that any expansion you buy on Xbox One will not transfer over to Xbox 360.
So if you are thinking on playing on both consoles you will have to pay twice for The Taken King.
For more information about the Digital Upgrade Program, go here:
https://www.bungie.net/en/Help/Article/12122
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Alright, thank you!