Answer
$\frac{-i}{1+i}$
= $- \frac{1}{2} - \frac{1}{2} i$
Work Step by Step
Now, $-i$ is at $270^\circ$ with unit 1, and
$1+i$ is at $45^\circ$ with absolute value $\sqrt{1^2 + 1^2} = \sqrt{2}$
Therefore, $\frac{-i}{1+i}$
= $\frac{1 cis270^\circ}{\sqrt{2}cis45^\circ}$
= $\frac{1}{\sqrt{2}}\cdot cis(270^\circ - 45^\circ)$ (Quotient Theorem)
= $\frac{1}{\sqrt{2}}\cdot cis225^\circ$
= $\frac{1}{\sqrt{2}}\cdot (\frac{-1}{\sqrt{2}} + \frac{-1}{\sqrt{2}} i)$
= $- \frac{1}{2} - \frac{1}{2} i$
