No you should not. HTTP explicitly suggests to send both headers:
the preferred behavior for an HTTP/1.1 origin server is to send both a strong entity tag and a Last-Modified value.
ETag
and Last-Modified
have a very similar purpose, but the semantic is different. With ETag you identify an entity, with Last-Modified you identify the date of a resource with a resolution of 1 second. Also the client behaviour is different:
- MUST use that entity tag
- SHOULD use Last-Modified
- SHOULD use both