Spotify refuses to officially add Chromecast support despite a lot of demand. There are several workarounds but they require installing more apps.
Here is how to cast Spotify on Android without installing anything extra.
With the introduction of Android OS 4.4.2, Google added native Chromecast Screencasting support to Android. While you're on the same WiFi network at your Chromecast, pull down your notification drawer and enable it.
Now open up Spotify and begind playing whatever music you'd like! The music will play through your TV.
You can even turn off your phone and the music will continue to play through the TV.
It's worth noting this works for all the major streaming apps: Pandora, Google Music, Soundcloud, Rdio, etc.
If you'd like to still use your phone while casting and if you've got a dedicated receiver, one trick is to plug the Chromecast into it instead of the TV. Set your receiver to only play the audio (and throw away the video).
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