Is there any way to simplify down this complex statement?

Question

The statement is;

$$\frac{1+e^{ix}}{1+e^{-ix}}$$

How would you multiply this equation so that you would get it to be simpler? I tried to multiply by $$\frac{e^{ix}}{e^{ix}}$$, but this only cancelled down the bottom, and I do not know what to do with the top! Please help, thank you!

Answer 1

Factor $e^{ix}$ in the numerator.

Note that $$ \frac{1+e^{ix}}{1+e^{-ix}}=\frac{e^{ix}(e^{-ix}+1)}{1+e^{-ix}} = e^{ix}$$

Answer 2

Hint: $$\frac{1+e^{ix}}{1+e^{-ix}}=\frac{e^{ix/2}(e^{-ix/2}+e^{ix/2})}{e^{-ix/2}(e^{ix/2}+e^{-ix/2})}$$