I don’t know about using a cable but a chromecast works fine - you can just cast your entire screen to it.
However, I find that sometimes the picture on the TV pauses as the wifi struggles to keep up (The workout on the phone carries on just fine). Downloading a workout beforehand helps reduce the network traffic though so it normally works ok.
Yes similar to Sandy… Chrome cast does cast to TV ok but for me it plays havoc with my Wahoo HRM ( drops out all the time and finally never comes back on)so much so that I have stopped using it and just do the workouts on my phone until I can come up with a better option