What will be the articles in the gaps?
He is both __ singer and __ dancer.
MY ASSUMPTION: a, ×.
I chose to write × for the second gap because, it indicates the same person.
You can find the noun dog in a sentence, for example, but you don't know which canine the writer means until you consider the entire noun phrase: that dog, Aunt Audrey's dog, the dog on the sofa, the neighbor's dog that chases our cat, the dog digging in the new flower bed. Modifiers can come before or after the noun. Ones that come before might include articles, possessive nouns, possessive pronouns, adjectives, and/or participles.
"We" is the subject pronoun and "singer" is the noun, the articles in this noun phrase should be used to assign the nouns to the subject pronoun, the use of the adjective "both" implies they should be grouped and so I would use "a singer and dancer":
He is both a singer and a dancer. - Grammatically correct, separates singer and dancer. This structure would be more suited to "he is
both an artist and a scientist", two disparate nouns.
He is both a singer and dancer. - "He" is the subject pronoun. The use of the word "both", means that both simultaneously apply. Therefore "singer and dancer" is an appropriate grouping and better in this context.
When we use both..and, we are using parallelling to attach the first part "He is.." to the both-clause and the and-clause. Whatever you do, it is important that these two clauses are the same.
If you wanted to talk about a particular person's skills, then an indefinite article or zero article are OK. Zero article makes the sentence sound quite literary: here is a typical example
That Jackson was both singer and dancer was evident during the Hollywood Palace broadcast. Performance and Popular Music: History, Place and Time
Using an indefinite article is more straightforward: it is less common in writing but may be more common in spoken English.
He is both a singer and a dancer.
If this person is the only member of a cast for a show that can sing and dance, you could also say
He is both the singer and the dancer