Calculate complex number considering...

Question

How can I calculate:

$$ \frac{1-Z}{1+Z} $$

...considering $Z = \cos(\alpha) + i \sin (\alpha)$

I have replaced the expression but I don't know how can I continue...

Answer 1

As given, $\;|z|=1\;$ ,so

$$\frac{1-z}{1+z}\cdot\frac{1+\overline z}{1+\overline z}=\frac{1+\overline z-z-|z|^2}{|1+z|^2}=-\frac{2\;\text{Im}\,z}{|1+z|^2}i$$

And since

$$|1+z|^2=|1+\cos x+i\sin x|^2=1+2\cos x+\cos^2x+\sin^2x=2(1+\cos x)\;,\;\;\text{Im}\,z=\sin x$$

we get

$$-\frac{2\sin x}{2(1+\cos x)}i=-\frac{\sin x}{2\left(1-\sin^2\frac x2\right)}i$$

The last identity just in case you want an expression with thesame trigonometric function. tThe denominator can be changed to $\;2\cos^2\frac x2\;$ , of course.