= Solution
For any real <orthonormal basis> $(G_i)$, the argument of (b)(ii) gives <almost sure convergence> and <convergence in L2> of $\widetilde Wh$. Finite linear combinations have <normal distributions>, and the <Parseval identity for a Hilbertian basis> gives
$$
\mathbb E[(\widetilde Wf)(\widetilde Wg)]=\sum_i\langle f,G_i\rangle\langle g,G_i\rangle
=\langle f,g\rangle.
$$
Thus $W$ and $\widetilde W$ are centered <Gaussian processes> with the same <covariance function>. More explicitly, their finite-dimensional <characteristic functions> are both
$$
\boxed{\mathbb E\exp\!\left(i\sum_{j=1}^mu_jWh_j\right)
=\exp\!\left(-\frac12\left\|\sum_{j=1}^mu_jh_j\right\|_{\mathcal H}^2\right).}
$$
This proves the <basis invariance of an isonormal Gaussian process>: equality of all <finite-dimensional distributions>, and hence equality in law of the indexed collections.
For the specified cosine <orthonormal basis>, put $\lambda_i=\pi(i-\tfrac12)$. Its coefficient against an interval indicator is
$$
\langle\mathbf1_{[0,t]},G_i\rangle
=\sqrt2\int_0^t\cos(\lambda_i s)\,ds
=\frac{\sqrt2\sin(\lambda_i t)}{\lambda_i}.
$$
Consequently the <Brownian half-integer sine expansion>, a <Karhunen–Loève expansion>, is
$$
\boxed{\widetilde B_t
=\sqrt2\sum_{i=1}^\infty b_i\frac{\sin(\pi(i-\tfrac12)t)}{\pi(i-\tfrac12)}.}
$$
For every fixed $t$, the sum converges <almost surely> and in <L2 space> because the coefficient squares sum to $\|\mathbf1_{[0,t]}\|^2=t$. Its <continuous modification> is a standard <Brownian motion> by (b)(v). The expansion also holds in $L^2(\Omega\times[0,1])$: the <expectation> of the squared tail is the integral of the coefficient-square tail, which tends to zero by the <dominated convergence theorem>, with bound $t$.
To realize this expansion for the original $B$, choose $b_i=WG_i$. These are <Jointly Gaussian> with <covariance matrix> equal to the identity, hence are <independent> <standard normal random variables>. The finite <orthogonal projections> of $\mathbf1_{[0,t]}$ onto $G_1,\ldots,G_N$ converge in $\mathcal H$, and the identity $\mathbb E|Wh|^2=\|h\|^2$ transfers that convergence to the displayed expansion for $B_t$. With an independently chosen sequence $(b_i)$ the resulting <Brownian motion> has the same law; a pathwise identification uses the coupling just given.
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