Yes
If they both have HDMI or component or composite video.
With HDMI the player will set itself to the resolution that the projector supports.
With component video if the output is higher that the projector can handle you won't get a picture. Connect the player via composite video to the projector, set the resolution to the native resolution of the projector. Reconnect the component video cables.
With composite video you are limited to 480p but you won't need to set anything to get a picture.
It has nothing to do with what was purchased when as much as what the models are to determine the input and output types. A brand new home theatre audio system might not have the right inputs.
With the information you provided (or more, what you did not provide), there is no simple yes or no. Give us the model numbers of your projector and Blu-Ray player and then we can give you a simple yes or no.
Yes
If they both have HDMI or component or composite video.
With HDMI the player will set itself to the resolution that the projector supports.
With component video if the output is higher that the projector can handle you won't get a picture. Connect the player via composite video to the projector, set the resolution to the native resolution of the projector. Reconnect the component video cables.
With composite video you are limited to 480p but you won't need to set anything to get a picture.