Yes, however if you want to use all of your transferred content without signing in to Xbox LIVE or with a different gamertag, you'll need to transfer the content licenses to your new console. Here’s how:
Sign in to Xbox LIVE using the gamertag that you used to purchase the content.
Go to Settings, and then select Account.
Go to Your Billing Options, and then select License Transfer.
Follow the on-screen instructions to transfer the content licenses.