Is there any difference between using this sentence and "요리사가 잘 요리해요"?
Well, that still makes sense though, but 요리사가 요리를 잘해요 is way much frequently used.
I'd also would write it like that but we should wait if someone knows it better
Would this phrase actually be used? Instead of something like "He is a good cook."
Then it should be “그는 좋은 요리사입니다“. And i think they will noy except it here in Duolingo!