Proof of a conjecture involving derangements and roots of unity

Let $n>1$ be an odd integer. For any primitive $n$-th root $\zeta$ of unity in the complex field. Via the Engenvector-eigenvalue Identity, we show that $$\sum _{\tau \in D(n-1)}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(\tau )\prod _{j=1}^{n-1}\frac{1+{\zeta }^{j-\tau (j)}}{1-{\zeta }^{j-\tau (j)}}=(-1{)}^{\frac{n-1}{2}}\frac{((n-2)!!{)}^2}{n},$$ where $D(n-1)$ is the set of all derangements of $1,\dots ,n-1$. This confirms a previous conjecture of Z.-W. Sun. Moreover, for each $\delta =0,1$ we determine the value of $\det [x+m_{jk}{]}_{1\le j,k\le n}$ completely, where $$m_{jk}=\{\begin{array}{ll}(1+{\zeta }^{j-k})\mathrm{/}(1-{\zeta }^{j-k})& \text{if }j≠k,\\ \delta & \text{if }j=k\mathrm{.}\end{array}$$