= Solution
For a deterministic <partition of an interval> $0=t_0<\cdots<t_m=t$, put $Q_\pi=\sum_j\Delta_jB\,\Delta_j\widetilde B$. <Independence> of the two <Brownian motions> and their <independent increments> gives
$$
\mathbb EQ_\pi=0,\qquad\mathbb EQ_\pi^2=\sum_j(t_j-t_{j-1})^2\leq t|\pi|\longrightarrow0.
$$
The defining sums for <quadratic covariation> therefore converge to zero in the <L2 space> and in <probability>. Hence, with the continuous versions of the <quadratic covariation>,
$$
\boxed{\langle B,\widetilde B\rangle_t=0\quad(t\geq0).}
$$
Equality holds simultaneously for all $t$ by checking rational times and using continuity. For <continuous local martingales>, the angle-bracket notation here agrees with <quadratic covariation>.
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