By what trigonometric trick does
\begin{align} \sin\alpha\Bigg[\cos(\omega t + \varphi)+\frac{\cos\alpha\sin(\omega t + \varphi)}{\sin\alpha}-\bigg(\cos(\varphi)+\frac{\cos\alpha\sin(\varphi)}{\sin\alpha}\bigg)e^{-\omega t/x}\Bigg]\\ \end{align}
reduce to
\begin{align} \bigg[\sin(\alpha+\omega t + \varphi)-\sin(\alpha+\varphi)e^{-\omega t/x}\bigg]? \end{align}
I've confirmed with Wolfram Alpha that this reduction is indeed correct.