Google announced a number of new updates coming to the Chromecast at I/O recently. One of the best is already rolling out: full device streaming.
The official Chromecast app has been updated to allow streaming your device's screen to your Chromecast. Initially the list of devices is limited to the following:
Nexus 4
Nexus 5
Nexus 7 (2013)
Nexus 10
Samsung Galaxy S4
Samsung Galaxy S5
Samsung Galaxy Note 3
Samsung Galaxy Note 10 (2014)
HTC One M7
LG G Pro2
LG G2
and LG G3
The update appears to be server-side, meaning you shouldn't need to update the app to see the new option. However, it's rolling out slowly so if you don't see it on one of the supported devices, sit tight.
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