Answer
$z_{1}z_{2}=20\displaystyle \sqrt{2}(\cos\frac{\pi}{4}+i\sin\frac{\pi}{4}),$
$z_{1}/z_{2}=\displaystyle \frac{5\sqrt{2}}{4}(\cos\frac{\pi}{4}+i\sin\frac{\pi}{4})$,
$1/z_{1}=\displaystyle \frac{1}{5\sqrt{2}}(\cos(-\frac{\pi}{4})+i\sin(-\frac{\pi}{4}))$
Work Step by Step
$z_{1}z_{2}=r_{1}r_{2}[\cos(\theta_{1}+\theta_{2})+i\sin(\theta_{1}+\theta_{2})]$
$\displaystyle \frac{z_{1}}{z_{2}}=\frac{r_{1}}{r_{2}}[\cos(\theta_{1}-\theta_{2})+i\sin(\theta_{1}-\theta_{2})],\quad z_{2}\neq 0$
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$z_{1}=5+5i$,
$\theta_{1}$ terminates in quadrant I
$\displaystyle \tan\theta_{1}=\frac{5}{5}=\mathrm{l} \Rightarrow$ $\displaystyle \theta_{1}=\frac{\pi}{4}$,
$r_{1}=\sqrt{25+25}=5\sqrt{2}$
$z_{1}=5\displaystyle \sqrt{2}(\cos\frac{\pi}{4}+i\sin\frac{\pi}{4})$
$z_{2}=4$,
$\theta_{2}=0$,
$r_{2}=4$.
$z_{2}=4(\cos 0+i\sin 0)$.
$z_{1}z_{2}=5\displaystyle \sqrt{2}\cdot 4[\cos(\frac{\pi}{4}+0)+i\sin(\frac{\pi}{4}+0)]$
$=20\displaystyle \sqrt{2}(\cos\frac{\pi}{4}+i\sin\frac{\pi}{4}),$
$z_{1}/z_{2}=\displaystyle \frac{5\sqrt{2}}{4}(\cos\frac{\pi}{4}+i\sin\frac{\pi}{4})$,
$1/z_{1}=\displaystyle \frac{1}{5\sqrt{2}}(\cos(-\frac{\pi}{4})+i\sin(-\frac{\pi}{4}))$