You are talking about an inverter; not an adapter.
An adapter changes the connection shape but does not change the fundamental nature of the signal.
In this case you need to take 12 volts DC (Direct - steady 12 volts - Current) and change it to Alternating - positive and then negative voltage 60 times a second - Current (AC) and THEN boost the 12 volts AC ten times to 120 volts AC and output that to a Duplex Outlet (standard house outlet).
The problem you will run into is the available 12 volt current at that "Cigarette Lighter" outlet. 8 Amps is what this outlet is fused for.
8 amps times 12 volts is 96 Watts. That is the maximum wattage that can be delivered without blowing the fuse. You can not upgrade the fuse without upgrading the wiring.
Check your TV for how many Watts is needs to operate and either buy a 12 volt DC TV (with less than 96 Watt draw) or buy a larger inverter (rated for at least the wattage of your TV); attach it directly to the battery per the directions in the inverter and as close to the battery as possible; then run an extension cord to your current TV.
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