Simplifying Difference Quotient

I was watching lecture 3 from MIT 1801 on single variable calculus, and I am having trouble understanding how the professor simplified the expression for the difference quotient. The original expression for the difference quotient was: $$\frac{\sin(x + \Delta x) - \sin(x)}{\Delta x}$$

and I understand how he simplified it to: $$\frac{\sin(x)\cos(\Delta x) + \cos(x)\sin(\Delta x) - \sin(x)}{\Delta x}$$ I just can't understand the simplification from the above expression to this: $$\sin(x)\biggl (\frac{\cos(\Delta x) - 1}{\Delta x}\biggr) + \cos(x)\biggl(\frac{\sin(\Delta x)}{\Delta x}\biggr)$$

Could someone explain the steps involved to get to this solution? Thanks,

• Factor a $\sin(x)$ out of the first and third terms in the numerator. – N. F. Taussig Oct 31 '17 at 17:02