Explanation

Sometimes, the more is the phrase short, the more it is tricky.

Analyzing the sentence overall is the first step to conquer it. The most important thing of your strategy is to grasp the very essence and the meaning of the sentence as a whole. Then this could be coupled with the searching of clues. These two approaches can be overlapped, using in conjunction, or you can use them as your convenience.

The first part of the sentence

Since she believed him to be both candid and trustworthy means that the first half our sentence will be a reason for the second half of the sentence.
Moreover, if she thinks that he is true (in the sense that we can trust him and does not lie) then the second part of the sentence, which contains the blank, must have as word something that expresses this sentiment. She might refuse to believe that it is dishonest.

Now, looking to the answer choices:

Irrelevant means unimportant, which clearly is not the right choice.

Facetious means sarcastic, who lies intentionally or using irony to mock someone else. it could be tempting but we need a word that shows us trust, loyalty or something else in a similar fashion.

Mistaken means, eventually wrong. However, you can be mistaken with or without being trustworthy.

Critical means something of extreme importance (usually) or even something that expresses disapproval or judgment. Clearly, it is a wrong answer.

Insincere means that someone when is insincere he/shge is not frank.

she refused to consider the possibility that his statement had been is not trustworthy or insincere

E is the best answer.

Regards

Attached is an image that should help.

Best of luck on your GRE and beyond,

-Brian



Source: Vince and Brian's GRE PowerPrep Explanations
_________________

Since she believed him to be both candid and trustworthy, she refused to consider the possibility that his statement had been (i) ______.


The first part of the sentence starting with "since" gives the reason for her refusal to consider the possibility that his statement could be something. Since she believed him to be both candid and trustworthy, she would have expected his statements to be sincere. Or in other words, she would not have believed that "his statement had been insincere". The word from the choice list is insincere.


_________________

The woman is dealing with someone she thinks is a saint (he/she is candid and trustworthy in her estimation). As such, the last thing she would consider is the person saying things that are not sincere. Insincere is thus the correct answer choice.