Step 1: Understand the sentence.
The sentence talks about the defense proving that an incident really took place, while the prosecution rejects it. Hence, the blanks must reflect (i) the manner of rejection and (ii) the reason given for rejection.
Step 2: Analyze the first blank (manner of rejection).
- Blithely = in a casual, cheerful manner — does not fit legal rejection.
- Cavalierly = dismissively, without proper seriousness — perfectly fits how the prosecution might reject something.
- Vehemently = strongly — suggests intensity, not dismissiveness.
- Persuasively = convincingly — but prosecution isn’t persuading, it’s rejecting.
- Convincingly = same issue, doesn’t fit.
Thus, the best fit for the first blank is cavalierly.
Step 3: Analyze the second blank (nature of rejection).
- Undesirable = unwanted — doesn’t match legal terminology.
- Apocryphal = of doubtful authenticity — exactly how the prosecution might describe the incident.
- Factual = the opposite of what prosecution claims.
- Pointless = irrelevant, doesn’t fit.
- Inevitable = makes no sense here.
Thus, the second blank should be apocryphal.
Step 4: Final sentence with correct option.
The defense proposes to show that the incident that the prosecution so cavalierly rejects as apocryphal did indeed take place.
\[
\boxed{\text{Correct Answer: (B) cavalierly, apocryphal}}
\]