The PC version of Tony Hawk's Pro Skater 4 only officially supports 4:3 fullscreen mode; the console versions, however, had 16:9 support. This was an in-game menu option in the PlayStation 2 and GameCube versions, but on the Xbox version, it was a system-level setting. Since the PC version was ported from the Xbox versions, the menu option is not there; however, with some hex editing, it is possible to access it.

Note that this does not change the resolution of the game; the game will still only run in a resolution of 640x480, 800x600, 1024x768, or 1280x1024. To run the game in widescreen mode, you will need a 16:9 monitor that can stretch an image from one of those resolutions to 16:9.

Also note that this only changes the horizontal FOV. On-screen text, HUD elements, menus and FMVs will appear stretched. This is also the case in the PlayStation 2 and GameCube versions.

Finally, note that this will result in you being unable to access the "Adjust Gamma" menu.

Open the C:\Program Files (x86)\Aspyr\Tony Hawks Pro Skater 4\Game\data\scripts\ folder in your THPS4 install folder.
There should be a gamemenu.qb file in this folder. WARNING: Back this file up before making any changes.
Open the file up in a hex editor such as XVI32.
Find the hex string 41646A7573742047616D6D61 ("Adjust Gamma")
Shortly before that, there should be an occurrence of the hex string 8A83D786. Replace it with 4901414C.
Find the first of two occurrences of the hex string 5374616E6461726420343A33 ("Standard 4:3").
Shortly before that, there should be an occurrence of the hex string 4901414C. Replace it with 8A83D786.
Launch THPS4 and enter any level (except practice mode).
Pause the game and select "Options", then "Screen Mode", then "Widescreen 16:9".
You will need to repeat the last two steps every time you launch the game.

A new way to make coexist both adjust gamma and widescreen option is possibly explained here

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