 Solution

Put $\mathcal{T}=\beta \hslash$ and $x=\omega \mathcal{T}$. With equal endpoints $q$, the Euclidean classical path and its action areWrite $q({\tau }_E)=q_{\mathrm{cl}}+\eta$, where $\eta (0)=\eta (\mathcal{T})=0$. Sine modes have fluctuation eigenvalues $(n\pi /\mathcal{T}{)}^2+{\omega }^2$. Relative to the free-particle determinant, their Gaussian integral contributesMultiplying the free prefactor $\sqrt{m/(2\pi \hslash \mathcal{T})}$ gives the Euclidean oscillator diagonal kernelIntegrating this over the common endpoint is precisely the trace. All fluctuations here are Dirichlet fluctuations about a fixed-endpoint path; periodicity is imposed on the full trace after endpoint integration.