An exact cause of death vs the exact cause of death

1.) It can be difficult to determine an exact cause of death.

2.) It can be difficult to determine the exact cause of death.

In respect to an / the exact cause of death:

1.) seems to represent any one exact cause of death, in a random case of death;

2.), on the other hand, appears to generally encompass individual cases involving death. I somehow senses that in terms of the function of "the," "cause" in 2.) plays a similar role as in "number" in the following expression "the number of deaths" or "pace" in "the pace of deaths."

Can you explain in as much detail as possible any subtle difference in meaning between the two sentences, and what the use of "a" or "the" might imply beyond the surface? In addition, please provide grammatical explanation, if possible.

There can be (and often are) multiple contributing causes of death.

The indefinite pronoun is preferred for statements or questions regarding one cause of death of many (or of an unknown number).

The definite pronoun is preferred for statements or questions regarding one cause of death when it is known or to be implied that that cause of death is the only cause of death.

In the question above, a generalization is being put forward. Because of this, it would be better to use the indefinite pronoun since the case is hypothetical in nature and the number of causes of death for that case is undefined (and therefore unknown).