How to simplify the ratio $1 - \cos2x + i\sin2x \over 1 + \cos2x - i\sin 2x$

Question

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 1

HINT

Recall that

• $\cos t = \frac{e^{it}+e^{-it}}{2}$

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

Answer 2

$$F=\frac{1-\cos 2x +i \sin 2x}{1+\cos 2x-i\sin 2x} = \frac{2 \sin ^2 x+ 2i \sin x \cos x}{2 \cos^2 x-2 i \sin x \cos x} =i \tan x \frac{\cos x -i \sin x}{\cos x -i \sin x}= i \tan x.$$

Answer 3

Using Intuition behind euler's formula

$$\dfrac{1-e^{-2ix}}{1+e^{2ix}}=\dfrac{e^{-ix}}{e^{ix}}\dfrac{e^{ix}-e^{-ix}}{?}=e^{2ix}\cdot\dfrac{2i\sin x}{2\cos x}=?$$