Yes, a full shut down, not just logging out.
Here’s how to reset the NVRAM on your particular MBP: Your Mac uses Non-Volatile Random-Access Memory (NVRAM). Before following the steps below, ensure you have no external hardware connected and make sure you use the built-in keyboard. See the dimensions: 1920 x 1080.I know you've already reset the NVRAM (what you're referring to as PRAM), but the first thing I'd try is to reset both the NVRAM and SMC in that order. To confirm the screen resolution on the external display (HDTV), I just took a screenshot of the entire desktop (Shift+Command+3). (At this time, MacBook just displays the desktop wallpaper). TIP: If you’re controlling the external display using MacBook only, then it’s recommended to turn its screen brightness and backlit keyboard light down to 0% to save the battery. MacBook Pro display working in full-screen on HDTV For controls, you need to use MacBook or you can attach a wireless keyboard and mouse for more comfort. That’s it! Now you’re ready to use your HDTV as an external display to watch movies, photo slideshow, etc.
Displaying 1080p (1920×1080) resolution in full-screen mode when connecting MacBook Pro with HDTV using an HDMI cable –.Now sound would just come from your Television set.
To enable Audio from TV’s speakers and NOT from MacBook Internal speakers – Open System Preferences > Sound, tap the Output tab and select your TV as a device for sound output.Select the HDMI port to which you’ve connected your MacBook Pro. Use the TV remote to switch to the Input device.We used the below one which is a single 6 feet cable that supports both audio and video, you can buy it from Amazon for $11.69 only. Connect both the devices using a quality Mini Display Port to HDMI Adapter Cable.How to connect MacBook Pro to TV with HDMI If you see “HDMI/ DisplayPort Output”, then your MacBook Pro does support HDMI audio out. Under Hardware, click “Audio (Built In)”. Click “About This Mac” and then “More Info…”, click System Report. Note: Only MacBook Pros April 2010 and later support audio, but if you want to check to know for sure, click the Apple logo in the top left corner.