It is about finding out this;
Google seach how to take a screen shot on you device with the buttons of the device....
(For the Samsung Galaxy s4 it is simultaneously holding power button @ top right, AND home button @ bottom center...for many devices that works, NOT FOR ALL of them)
Then try taking a screenshot...works? (Verify in gallery that it is there)
Then go to the app you would like to use with chromecast...take a screen shot.
If that works then nothing prevents it to be displayed with the chromecast on a TV....IF.
Your device supports "Miracast" or "Screen Mirrorring" functionality.
So requirements are;
1. The ability to take a screen shot of the app in use you would like to display on TV with chromecast.
2. The device used to take that above/fore-mentioned Screenshot needs to support "Miracast" or "Screen Mirrorring" functionality.
If the here directly above paragraph conditions are met, it will work and functioning of the desired is granted.
Explanatory; if a Screenshot can be taken that means that there is no DRM content protection in that app...
Meaning there is nothing prohibiting the Scene Mirroring functionality to display it on a display or TV.
Try it with a banking app...or Netflix, it will not allow Screenshots because the content is DRM protected.
DRM protected content can ONLY be streamed and in such case "casted" and not true mirroring functionality to prevent copies being made....IF someone manages succesfully to take a screen shot OF a app provided by your bank WHILE sensitive info OF ANY KIND is displayed CONTACT YOUR BANK AND EXPLAIN THAT THERE APP IS NOT MEETING CONTENT AND PRIVACY PROTECTION REQUIREMENTS. (Web browsers not included!!!)
For that matter....an app provided by a bank should never allow sceenshots.