<s>[INST] The roots of the equation $z^n=(1+z)^n$... The complex roots of the equation 

$$z^{n}=(1+z)^{n}$$

$A.$ are vertices of a regular polygon
$B.$ lie on a circle
$C.$ are collinear
$D.$ none of these
Don't know where to start.. Please help.!
NOTE:  $z$ is a complex number.
 [/INST] Clearly, $z\ne0,$
So dividing either sides by $z^n,$ we have $$\left(1+\frac1z\right)^n=1$$
Using  de Moivre's formula & Euler's formula, $$1+\frac1z=e^{\frac{2m\pi i}n}=\cos\frac{2m\pi}n+i\sin\frac{2m\pi }n$$ where $m$ is any integer
$$\implies \frac1z=\cos\frac{2m\pi}n+i\sin\frac{2m\pi }n-1=i2\sin\frac{m\pi}n\cos\frac{m\pi}n-2\sin^2\frac{m\pi}n$$
$$=2i\sin \frac{m\pi}n\left(\cos\frac{m\pi}n+i\sin \frac{m\pi}n\right)$$ (using $\cos2A=1-2\sin^2A$)
$$z=\frac1{2i\sin \frac{m\pi}n\left(\cos\frac{m\pi}n+i\sin \frac{m\pi}n\right)}=\frac{{\cos\frac{m\pi}n-i\sin \frac{m\pi}n}}{2i\sin \frac{m\pi}n}=-\frac12-i\frac{\cot\frac{m\pi}n}2$$
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