It's good that you are considering your TV capibilities before purchasing a TT, but instead of asking "Can my TV handle it"?, ask instead how to figure it out for yourself, and then do so. Everyone on here is very helpful, but don't go on assumptions made with too little info. Looking up the numbers on a similiar TV to your 2013 Tundra, I see the following:
Towing Capacity: 9000 lbs. Payload Capacity: 1575 lbs. Gross Vehicle Weight: 7200 lbs.
You need to refer to your manual and the door sticker on yours as these can vary depending on equipment. Hopefully you also have the tow package. Remember, that payload is used up way before towing capacity is met. Payload includes the stuff you have in the TV - all the passangers, tank of gas, stuff in the bed, weight of the hitch receiver, and toung weight of the TT.