trigonometry - How to simplify the ratio $1 - cos2x + isin2x over 1 + cos2x - isin 2x$

The ratio is as follows:

$$1 - \cos2x + i\sin2x \over 1 + \cos2x - i\sin 2x$$

I am unsure how to simplify this, as the numerator poses a problem as I try to multiply this equation by $\operatorname{cis}(2x)$ to get a real denominator.

Answer

HINT

Recall that

• $\cos t = \frac{e^{it}+e^{-it}}{2}$




• $\sin t = \frac{e^{it}-e^{-it}}{2i}$