Does anyone know how to show that the stochastic integrals
\begin{equation} \bigg\{ \int_0^1 \cos \Big[ (n- \frac{1}{2}) \pi t \Big] \,dW_t \bigg\}_{n \in \mathbb{N}} \end{equation}
are independent. Do we need to consider the joint characteristic function
$$ \mathbb{E} \bigg[ \exp \bigg\{ i \bigg( \alpha_1 \int_0^1 cos \Big[ (n_1 - \frac{1}{2}) \pi t \Big] \,dW_t + \alpha_2 \int_0^1 cos \Big[ (n_2 - \frac{1}{2}) \pi t \Big] \,dW_t \bigg) \bigg\} \bigg] \quad ?$$
For notational convenience, define $f_{n,t}:=\cos(n-1/2)\pi t$ and choose any finite set of $k$-distinct natural numbers $n_1, \ldots, n_k$. Then, the Itô integrals $\{\int_0^1 f_{n_1,t}~\text dW_t,\ \ldots,\ \int_0^1 f_{n_k,t}$$\text dW_t\}$ are jointly normal. Consequently, these intergrals are independent if any pair of integrals has zero correlation. So, for $i,j=1\ldots k$, we could proceed to show this as follows:
$$ \begin{array}{rcl} &\mathbb C\text{ov}\big[\int_0^1 f_{n_i,t}~\text dW_t,~\int_0^1 f_{n_j,s}~~\text dW_s\big] = \mathbb E\Big[ \int_0^1 f_{n_i,t}\text dW_t\int_0^1 f_{n_j,s}\text dW_s\Big]&\\ &=\mathbb E\Big[\int_0^1 f_{n_i,t}f_{n_j,t}\text d\langle W\rangle_t\Big]= \int_0^1\cos(n_i-1/2)\pi t~\cos(n_j-1/2)\pi t~\text dt&\\ &=\frac{1}{2}\delta_{ij},& \end{array} $$
where we have used the orthogonality of the functions $f_{n_i,t}$ for all $i$ over the integration range $0\leqslant t\leqslant 1$. We are done.
Remarks: As an aside, note that $\mathbb E[\int_0^1 f_{n,t}\text dW_t]=0$, for all $n$. So, following our preceding computation, we can now be more explicit in stating that the Itô integrals are each normally distributed as $\int_0^1 f_{n,t}\text dW_t \sim \mathcal N(0,\frac{1}{2})$. Moreover, each set of $k$ distinct integrals is jointly distributed as $\mathcal N(\vec{0}_k,\frac{1}{2}I_{k})$, where $\vec{0}_k$ is the $k$-dimensional zero vector and $I_{k}$ is the $k$-dimensional identity matrix.