The squared sine values are the roots of the cubic minimal polynomial x^3-7x^2+14x-7. Their power sums begin 3,7,21 and satisfy the recurrence S_n=7S_(n-1)-14S_(n-2)+7S_(n-3). Exact 7-adic valuations through n=24 follow from that recurrence. In each residue class modulo 3, every recurrence summand has valuation at least floor(n/3), so induction proves the divisibility theorem. This is exact algebraic computation, not independent verification of the original trigonometric derivation.
