exponential function and hyperbolic function inequality

Question

I try to solve the following inequality: Let $a>0$, $b\in\mathbb{R}$, and $t\geq 0$. Then we have \begin{equation} e^{\frac{-at}{2}}*\frac{\sinh(\frac{t\Delta}{2})}{\Delta}\leq Ce^{\frac{-t\cdot b^2}{a}} \end{equation} where \begin{equation} \frac{a}{2}\leq \Delta\leq a \ \ \ \text{and} \ \ \ \Delta=\sqrt{a^2-4b^2} \end{equation}