Determine all theta satisfying an expression

Question

For what values of θ does the following equation hold $$∏^{100}_{k=1} [\cos (kθ)+ i \sin (kθ)] = 1.$$

We can assume $∑^n_{i=1} i =\frac{n(n+1)}{2}$ for all natural numbers $n$.

Answer 1

By Euler's formula, $\displaystyle \prod_{k=1}^{100}e^{ik\theta} = 1 \rightarrow e^{i\theta+2i\theta+\cdots+100i\theta} = 1 = e$. Thus, $e^{5050i\theta} = 1 \rightarrow 5050\theta = 360^\circ\pi k$ and $\theta = \dfrac{36^\circ}{505}k$.