How would you simplify $\frac{ 2}{1+z}$, where $z$ is equal to $\text{cis }x$?

Question

I am struggling with this question as I am unsure of how to get rid of the complex denominator. Is there some trigonometric rule I need to use to crack this question? In the book, it is meant to equal; $(1-i) \cdot \tan \frac{x}{2}$, but how?

Answer 1

If you mean "what is $2/(1+z)$ where $z=\cos x+i\sin x=e^{ix}$?'' then $$\frac{2}{1+e^{ix}}=\frac{2e^{-ix/2}}{e^{-ix/2}+e^{ix/2}} =\frac{\cos(x/2)-i\sin(x/2)}{\cos(x/2)}=1-i\tan(x/2).$$ Here I used the identities $e^{it}+e^{-it}=2\cos t$ and $e^{-it} =\cos t-i\sin t$.