Answer
$\begin{align}
& \frac{\left[ 2\left( \cos \,60{}^\circ +i\,\sin \,60{}^\circ \right) \right]\left[ \sqrt{2}\left( \cos \,315{}^\circ +i\,\sin \,315{}^\circ \right) \right]}{\left[ 2\times 2\left( \cos \,330{}^\circ +i\,\sin \,330{}^\circ \right) \right]}; \\
& \frac{\sqrt{2}}{2}\left( \cos \,45{}^\circ +i\,\sin \,45{}^\circ \right);\ \approx \frac{1}{2}+\frac{i}{2} \\
\end{align}$
Work Step by Step
Consider the provided expression
$z=\frac{\left( 1+i\sqrt{3} \right)\left( 1-i \right)}{\left( 2\sqrt{3}-2i \right)}$
First convert it into polar form:
$\begin{align}
& z=\frac{\left( 1+i\sqrt{3} \right)\left( 1-i \right)}{\left( 2\sqrt{3}-2i \right)} \\
& =\frac{\left[ 2\left( \cos \,60{}^\circ +i\,\sin \,60{}^\circ \right) \right]\left[ \sqrt{2}\left( \cos \,315{}^\circ +i\,\sin \,315{}^\circ \right) \right]}{\left[ 2\times 2\left( \cos \,330{}^\circ +i\,\sin \,330{}^\circ \right) \right]}
\end{align}$
Apply the multiplication rule of complex numbers in the numerator of the above expression:
$z=\frac{\left( \sqrt{2}\times 2 \right)\left( \cos \left( 60{}^\circ +315{}^\circ \right)+i\,\sin \left( 60{}^\circ +315{}^\circ \right) \right)}{\left[ 2\times 2\left( \cos \,330{}^\circ +i\,\sin \,330{}^\circ \right) \right]}$
Simplify the above expression:
$z=\frac{\sqrt{2}\left( \cos \,375{}^\circ +i\,\sin \,375{}^\circ \right)}{2\left( \cos \,330{}^\circ +i\,\sin \,330{}^\circ \right)}$
Apply the quotient rule of complex numbers in the above expression:
$\begin{align}
& z=\frac{\sqrt{2}\left( \cos \left( 375{}^\circ -330{}^\circ \right)+i\,\sin \left( 375{}^\circ -330{}^\circ \right) \right)}{2} \\
& =\frac{\sqrt{2}\left( \cos \,45{}^\circ +i\,\sin \,45{}^\circ \right)}{2}
\end{align}$
The above expression is the polar form of the provided expression.
Convert it into rectangular form by substituting the values of $\cos \,45{}^\circ $ and $\sin \,45{}^\circ $ in the above expression:
$\begin{align}
& z=\frac{\sqrt{2}}{2}\left( \frac{1}{\sqrt{2}}+i\frac{1}{\sqrt{2}} \right) \\
& =\frac{\sqrt{2}}{2\sqrt{2}}\left( 1+i \right)
\end{align}$
Simplify the above equation:
$z=\left( \frac{1}{2}+\frac{i}{2} \right)$
The above expression is the rectangular form of the provided expression.
The polar form of the provided expression is $\frac{\sqrt{2}}{2}\left( \cos \,45{}^\circ +i\,\sin \,45{}^\circ \right)$.
The rectangular form of the provided expression is $\approx \frac{1}{2}+\frac{i}{2}$.
