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I am writing this question because I do not know if this setup would work. 

I would like to send the video signal from the Fire Stick TV (netflix content) to an HDMI to Optical Extrator and forward it to my projector (BenQ TH585) through the HDMI OUT. 

For the audio I would use the Optical Output from the extractor and connect it to my Sonos Beam. 

I would appreciate if someone could confirm if this setup would work.

Thanks in advance! 

If the Fire TV stick video-output works when it’s plugged directly into either the HDMI port or the HDMI-MHL input on the rear of the projector, then I don’t see a reason why the splitter/extractor wouldn’t work - you also will need to use the Optical to HDMI adapter that comes with your Beam.