I'm looking over some class notes, and I'm sure that my professor has made an error. Consider the following equation:

$$ e^{-q} \sin \left(kx - \omega{t} \right) - e^{q} \sin \left(kx + \omega{t} \right)$$

In my notes, this can be simplified as:

$$ e^{-q} \left(\sin\left(kx - \omega{t}\right) - \sin \left(kx + \omega{t}\right)\right)- \left(e^{q} - e^{-q}\right) \sin \left(kx - \omega{t}\right)$$

However, I'm not convinced this is correct. I thought that the simplification for the left hand terms should be

$$\sin\left(kx - \omega{t}\right) = \sin(kx)\cos(\omega{t}) - \cos(kx) \sin(\omega{t})$$

which is different from his solution. Also, I'm not sure what is happening with the terms on the right hand side. For example, why is $e^q$ simplified to $e^q - e^{-q}$

Any advice would be appreciated