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Use AirPlay on your Mac
Show whatever’s on your Mac mini on the big screen using AirPlay Mirroring. To mirror the Mac mini screen on your TV screen or to use the HDTV as a second display, connect your HDTV to Apple TV and make sure the Apple TV is on the same Wi-Fi network as your Mac mini. You can also play some web videos directly on your HDTV without showing what’s on your desktop—handy when you want to play a movie but keep your work private.

Apple TV is sold separately at Connect a display to Mac mini.