Reduce a trigonometric expression 2.0

Question

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.

Answer 1

Have you even bothered to look up any trig identities?

Using the formula \begin{align} \sin(x) \cos(y) = \frac{1}{2} [\sin(x+y) + \sin(x-y)] \end{align}

and the fact that $\sin$ is an odd function, you can verify this result in about 60 seconds. Perhaps you should at least attempt the problem before posting about it?