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For $\omega_j\in \left(\frac{(j-1)\pi}{1-R},\frac{j\pi}{1-R}\right)\;(j\geq 1)$, where $R\in (0,1)$, prove or disprove that $$ \mbox{sgn}\left(-\cos(\omega_j)-\omega_j\sin(\omega_j)+q(R-1)\cos[\omega_j(1+R)]\right)=\mbox{sgn}(q)\;, $$ where $q\in\mathbb{R}$.

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