Yes, those tools are cool...however, you may need the Spatial Analyst and a higher ArcGIS Desktop license to run any or all of the associated tools. For your requirements, Clip in Data Management (supported at all license levels) essentially does the 'Erase' function if you use the ClipGeometry option (the check box) and provide a geometry with a 'hole' in it - presumably that would be your 'smaller extent' you wish to remove.
Just have to be careful with the 'promoting pixel depth'...your output raster format has to be able to support that operation if that has effect...I used tif input along with the requisite polygon w/ a hole in it to mimic your requirement, and reset the raster properties on the added output in the map to not 'stretch' (to match symbology settings as in the original tif).
Someone may have already committed some tool 'magic' on the input ClipGeometry in the tool Gallery somewhere - sorry, I don't have time for it, and it's a simple thing really anyway to create it yourself. The main thing here is that the Clip tool supports 'erase' functionality...the tool(s) Steve posted far exceeds this functionality.
Enjoy,
Wayne
EDIT: So, actually Rodney had the 'answer' at post #4. (if you want to award him)