The complex number $z=\frac{i-1}{\cos \frac{\pi }{3}+i\sin \frac{\pi }{3}}$ is equal to:

Given,

$z=\frac{i-1}{\cos {\displaystyle \frac{\pi }{3}}+i\sin {\displaystyle \frac{\pi }{3}}}$

$\Rightarrow z=\frac{\sqrt{2}\left({\displaystyle \frac{-1}{\sqrt{2}}}+{\displaystyle \frac{i}{\sqrt{2}}}\right)}{\cos {\displaystyle \frac{\pi }{3}}+i\sin {\displaystyle \frac{\pi }{3}}}$

Now argument of complex number $\frac{-1}{\sqrt{2}}+\frac{i}{\sqrt{2}}$ is $\frac{3\mathrm{\pi }}{4}$

So, by euler's formula $e^{i\theta }=\cos \theta +i\sin \theta$ we get,

$\Rightarrow z=\frac{\sqrt{2}{e}^{i3\pi /4}}{e^{i\pi /3}}$

$\Rightarrow z=\sqrt{2}{e}^{i5\pi /12}$

$\Rightarrow z=\sqrt{2}\left(\cos \frac{5\pi }{12}+i\sin \frac{5\pi }{12}\right)$