Answer
$$ - \frac{1}{2} - \frac{1}{2}i$$
Work Step by Step
$$\eqalign{
& \frac{{ - i}}{{1 + i}} \cr
& {\text{Convert the numerator and the denominator to trigonometric form}} \cr
& - i = \left( {\cos {{270}^ \circ } + i\sin {{270}^ \circ }} \right) \cr
& 1 + i = \sqrt 2 \left( {\cos {{45}^ \circ } + \sin {{45}^ \circ }} \right) \cr
& {\text{Then,}} \cr
& \frac{{ - i}}{{1 + i}} = \frac{{\cos {{270}^ \circ } + i\sin {{270}^ \circ }}}{{\sqrt 2 \left( {\cos {{45}^ \circ } + \sin {{45}^ \circ }} \right)}} \cr
& {\text{Using the Quotient Theorem}} \cr
& = \frac{1}{{\sqrt 2 }}\left[ {\cos \left( {{{270}^ \circ } - {{45}^ \circ }} \right) + i\sin \left( {{{270}^ \circ } - {{45}^ \circ }} \right)} \right] \cr
& {\text{Simplify}} \cr
& = \frac{1}{{\sqrt 2 }}\left[ {\cos \left( {{{225}^ \circ }} \right) + i\sin \left( {{{225}^ \circ }} \right)} \right] \cr
& {\text{Write in Rectangular form}} \cr
& = \frac{1}{{\sqrt 2 }}\left( { - \frac{{\sqrt 2 }}{2} - \frac{{\sqrt 2 }}{2}i} \right) \cr
& = - \frac{1}{2} - \frac{1}{2}i \cr} $$
