Yes - here's several ways it can be done:
Chromecast method
There are several media steaming devices on the market which plug into the HDMI socket on your TV and connect to your wifi. Using one of these you can 'cast' directly from your computer to the TV. We've had success using devices like Amazon Fire, Google's Chromecast and Apple TV
YouTube sign in method
if you have a fancy Smart TV, you can use it to sign in on the YouTube app. Sign in on your laptop too, and click your private show link. Back on the TV, move to your history, and you should see the show listed at the top
Direct HDMI connection
Some laptops have a HDMI out socket on them. You can use this to connect directly to your TV