Given \(\left(B_{t}ight)_{t \in \mathbb{R}_{+}}\)a standard Brownian motion and \(n \geqslant 1\), let the random variable \(X_{n}\) be defined as

\[
X_{n}:=\int_{0}^{2 \pi} \sin (n t) d B_{t}, \quad n \geqslant 1 \]

a) Give the probability distribution of \(X_{n}\) for all \(n \geqslant 1\).

b) Show that \(\left(X_{n}ight)_{n \geqslant 1}\) is a sequence of identically distributed and pairwise independent random variables.
Hint: We have \(\sin a \sin b=\frac{1}{2}(\cos (a-b)-\cos (a+b)),

a, b \in \mathbb{R}\).