Explanation

The sentence focuses on the relationship between the gaps in existing accounts of a life and the evidence used to produce those accounts. Since “gaps” implies a concern with completeness, the characterization of that evidence that makes the most sense for Blank (ii) is “incomplete.” The other choices, “credible” or “extant,” could explain the accuracy or verifiability of the accounts in question but nothing about the gaps themselves. Once it is determined that “incomplete” is the best choice for characterizing the evidence, it follows that the gaps in the accounts based on that evidence would likely be considerable, so the opposite of considerable, “trifling,” is the correct answer for Blank (i). Thus the correct answer is trifling (Choice A) and incomplete (Choice F).
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Because trifling means meaningless, insignificant NOT important, trivial.

For this reason is the right one. You instead think that trifling means IMPORTANT. It is exactly the contrary.

Hope this helps

Regards
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Your reflection is not without sense.

However, in the second part of the sentence, the right word is incomplete BUT we do have gaps in the first part of the same. As such, the gaps are NOT trivial i.e they are important.

The gaps in existing accounts of the playwright’s life are not trifling, which means they are important eventhough theay are incomplete.

I admit is convoluted and tough the sentence.

Hope this helps.

Ask if something does not convince you yet.

Regards
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B and D

The definition of the word trifling is trivial or unimportant. Let's complete the first blank with one of the definitions of trifling.

The gaps in existing accounts of the playwright’s life are not UNIMPORTANT, since much of the documentary evidence on which historians have relied is (ii) ________.


The presence of a "not" in front of the unimportant creates a double negative (something that is NOT IMPORTANT would be unimportant and something that is NOT UNIMPORTANT would be important). This means that the first part of the sentence means that the gaps in the existing accounts are important. If the gaps are important then we need a word that justifies it in the second part. If the documentation were "credible" then nobody would be arguing there are gaps about the playwritghts' life. "Extant" may fit but it doesnt justify why there are gaps. "incomplete" works because incomplete historical documentation would lead to gaps in the understanding of a historical event.

I personally hate this question because English ia very direct language and that double negative is a little infuariating. People dont say "i am NOT doing unwell" when they want to say they are doing well.