Given $z = \text{cos}\alpha + i \text{sin} \alpha$ and $w = \text{cos}\beta + i \text{sin} \beta$

$z - w = \text{cos}\alpha + i \text{sin} \alpha -(\text{cos}\beta + i \text{sin} \beta)$

$z - w = \text{cos}\alpha -\text{cos}\beta + + i \text{sin} \alpha - i \text{sin} \beta)$

$z - w = - 2 \text{sin}(\frac{\alpha-\beta}{2})\text{sin}(\frac{\alpha+\beta}{2}) + i 2 \text{sin}(\frac{\alpha-\beta}{2})\text{cos}(\frac{\alpha+\beta}{2})$

$z - w = 2 i \text{sin}(\frac{\alpha-\beta}{2}) (\text{cos}(\frac{\alpha+\beta}{2}) + i \text{sin}(\frac{\alpha+\beta}{2}))$

$z - w = 2 i \text{sin}(\frac{\alpha-\beta}{2}) e^{\frac{\alpha + \beta}{2}}$