From what everyone has been saying and it really sounds like it's the truth, is that you can only have TTK available on the profile you purchased it with. So if you have more than one profile you play on you'll have to purchase another taken king dlc all together.
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That seems more like speculating, as Bungie never said anything about it.
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THIS IS TRUE. NO SPECULATION ABOUT IT. Because they never said anything about it I'm HOPING it is a glitch on the 360. I did purchase it on the Xbox One too and all content shares across all profiles fine. It just doesn't share to all profiles on the Xbox 360.
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I'm hoping it's a glitch, too....