Here the reading for this in the original manuscript is difficult to determine; the word looks most like a defective this, but the is also possible. Oliver Cowdery copied this word into 𝓟 as this. Also while copying, he initially wrote “vale of tears” in 𝓟; then somewhat later (with slightly heavier ink flow) he corrected tears to sorrow. His correction was probably made when he proofed 𝓟 against 𝓞. Although 𝓞 is not extant for the word sorrow or tears, it probably read “vale of sorrow” since either reading, “vale of tears” or “vale of sorrow”, is possible here, and consequently there would have been little tendency for Oliver to have consciously edited out one version in favor of the other. In fact, Literature Online lists numerous instances of “this vale of tears” and “this vale of sorrow”, but there are about 14 times more instances with tears than with sorrow. This difference in frequency probably explains why Oliver initially wrote “this vale of tears” in 𝓟. There are examples of “this vale of sorrow” dating from Early Modern English up to the present, as in these examples from Literature Online (with accidentals modernized here):
The Oxford English Dictionary also specifically notes that the determiner for this phrase is usually this (see definition 2b under vale). For that reason, it is probably best to reject the as the reading of the original manuscript in Alma 37:45.
Summary: Accept the idiomatic expression “this vale of sorrow” in Alma 37:45, the reading in 𝓟 (with initial tears corrected to sorrow).