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.