Let $Q_k(x)$ be Stanley's explicit denominator for the dimer-covering generating function $F_k(x)=\sum_{n\ge0}A_{k,n}x^n$ of $k\times n$ rectangles. Stanley conjectured in 1985 that $Q_k(x)$ has only simple roots; this longstanding conjecture was recently recorded in Lai's list of open problems on tilings (see [6, Problem 33]). We disprove the conjecture by proving that $Q_{14h-1}(x)$ and $Q_{30h-1}(x)$ have repeated roots for every $h\ge1$; in particular, $k=13$ is the smallest counterexample. The construction comes from two exceptional multiplicative identities among trigonometric algebraic units. We further propose a conjecture concerning this class of trigonometric identities, which appears to be related to Robinson's problem on primitive Pell factors.