Explanation

(c)
$(\cos{\theta + i\sin{\theta)({cos2}{\theta + i\sin{2\theta)\ldots(\cos{n\theta + i\sin{n\theta) = 1}}}}}}$
$\Rightarrow \ \cos{(\theta + 2\theta + 3\theta + ... + n\theta)} + \ i\sin{(\theta + 2\theta + 3\theta + ... + n\theta) = 1}$
$\Rightarrow \ \cos\left( \frac{n(n + 1)}{2}\theta \right) + i\sin{\left( \frac{n(n + 1)}{2}\theta \right) = 1}$
On comparing the coefficients of real and imaginary on both sides, we get
$\ \cos{\left( \frac{n(n + 1)}{2}\theta \right) = 1}$
and $\sin{\left( \frac{n(n + 1)}{2}\theta \right) = 0}$
$\Rightarrow \ \ \left( \frac{n(n + 1)}{2}\theta \right) = 2m\pi$
$\Rightarrow \ \$ $\theta = \frac{4m\pi}{n(n + 1)}$, where $m \in I$