  In a previous paper arXiv:2406.06294 [math.NT], the author proved the exact
formulae for ranks of partitions modulo each prime $p\geq 5$. In this paper,
for $p=5$ and $7$, we prove special vanishing properties of the Kloosterman
sums appearing in the exact formulae. These vanishing properties imply a new
proof of Dyson's rank conjectures. Specifically, we give a new proof of
Ramanujan's congruences $p(5n+4)\equiv 0\pmod 5$ and $p(7n+5)\equiv 0\pmod 7$.
