\begin{enumerate}
  \item Give the exponential and trigonometric forms of the complex numbers $1 + \mathrm{i}$ and $1 - \mathrm{i}$.
  \item For every natural number $n$, we define
$$S_{n} = (1 + \mathrm{i})^{n} + (1 - \mathrm{i})^{n}.$$
a. Determine the trigonometric form of $S_{n}$.\\
b. For each of the two following statements, say whether it is true or false by justifying your answer. An unjustified answer will not be taken into account and the absence of an answer is not penalised.

Statement A: For every natural number $n$, the complex number $S_{n}$ is a real number.\\
Statement B: There exist infinitely many natural numbers $n$ such that $S_{n} = 0$.
\end{enumerate}