Answer
$2\left( \cos 5{}^\circ +i\sin 5{}^\circ \right)$
Work Step by Step
Division of two complex numbers:
The division of two complex numbers ${{z}_{1}}={{r}_{1}}\left( \cos {{\theta }_{1}}+i\sin {{\theta }_{1}} \right)$ and ${{z}_{2}}={{r}_{2}}\left( \cos {{\theta }_{2}}+i\sin {{\theta }_{2}} \right)$, is given by
$\frac{{{z}_{1}}}{{{z}_{2}}}=\frac{{{r}_{1}}}{{{r}_{2}}}\left[ \cos \left( {{\theta }_{1}}-{{\theta }_{2}} \right)+i\sin \left( {{\theta }_{1}}-{{\theta }_{2}} \right) \right]$
So, the product of the given complex numbers ${{z}_{1}}=10\left( \cos 10{}^\circ +i\sin 10{}^\circ \right)\text{ and }{{z}_{2}}=5\left( \cos 5{}^\circ +i\sin 5{}^\circ \right)$ is
$\begin{align}
& \frac{{{z}_{1}}}{{{z}_{2}}}=\frac{10}{5}\left[ \cos \left( 10{}^\circ -5{}^\circ \right)+i\sin \left( 10{}^\circ -5{}^\circ \right) \right] \\
& =2\left( \cos 5{}^\circ +i\sin 5{}^\circ \right)
\end{align}$
Hence, the product of the complex numbers is $2\left( \cos 5{}^\circ +i\sin 5{}^\circ \right)$.
