Step 1: Understanding the Concept:
This question tests knowledge of a common fixed idiom rather than a general grammar rule.
Step 2: Key Formula or Approach:
Some phrases in English are fixed, meaning only one specific word slots correctly into them no matter how logical another word might sound. Recognizing the idiom "try as I might" as a whole unit is the fastest way to answer this kind of question, rather than testing each modal verb for meaning alone.
Step 3: Detailed Explanation:
The sentence structure "Try as I ___, [result did not happen]" is a concessive construction meaning "no matter how hard I tried, the result still did not happen".
This exact construction is idiomatically fixed with "might": "Try as I might, I couldn't open the door" is a natural, commonly used English sentence.
Swapping in "could", "should" or "would" breaks this fixed pattern. None of these three appear in standard usage in this construction, even though they are all modal verbs like "might".
Step 4: Final Answer:
The word that correctly completes the idiom is "might", so the answer is option (c).