Answer
$$ - \frac{{\sqrt 3 }}{2} - \frac{1}{2}i$$
Work Step by Step
$$\eqalign{
& \frac{{2i}}{{ - 1 - i\sqrt 3 }} \cr
& {\text{Convert the numerator and the denominator to trigonometric form}} \cr
& 2i = 2\left( {\cos {{90}^ \circ } + i\sin {{90}^ \circ }} \right) \cr
& - 1 - i\sqrt 3 = 2\left( {\cos {{240}^ \circ } + \sin {{240}^ \circ }} \right) \cr
& {\text{Then,}} \cr
& \frac{{2i}}{{ - 1 - i\sqrt 3 }} = \frac{{2\left( {\cos {{90}^ \circ } + i\sin {{90}^ \circ }} \right)}}{{2\left( {\cos {{240}^ \circ } + \sin {{240}^ \circ }} \right)}} \cr
& {\text{Using the Quotient Theorem}} \cr
& = \frac{2}{2}\left[ {\cos \left( {{{90}^ \circ } - {{240}^ \circ }} \right) + i\sin \left( {{{90}^ \circ } - {{240}^ \circ }} \right)} \right] \cr
& {\text{Simplify}} \cr
& = \cos \left( { - {{150}^ \circ }} \right) + i\sin \left( { - {{150}^ \circ }} \right) \cr
& = \cos \left( {{{150}^ \circ }} \right) - i\sin \left( {{{150}^ \circ }} \right) \cr
& {\text{Write in Rectangular form}} \cr
& = - \frac{{\sqrt 3 }}{2} - i\left( {\frac{1}{2}} \right) \cr
& = - \frac{{\sqrt 3 }}{2} - \frac{1}{2}i \cr} $$
