= Solution
Define the radial <martingale> and its clock by
$$
H_t=\int_0^tX_s\,dX_s+Y_s\,dY_s,\qquad\boxed{A(t)=\int_0^t(X_s^2+Y_s^2)\,ds.}
$$
The <Itô formula> gives $R_t^2=2H_t+2t$. Independence of the coordinate <Brownian motions> makes their cross variation zero, so
$$
\langle H\rangle_t=A(t),\qquad\langle Z\rangle_t=A(t),\qquad\langle H,Z\rangle_t=\int_0^t(X_sY_s-Y_sX_s)\,ds=0.
$$
The last identity is <orthogonality of the radial martingale and planar Brownian area>.
The clock is adapted and continuous, and it is strictly increasing almost surely. Otherwise the two coordinate paths would both vanish throughout a nontrivial interval. Such an interval contains a rational subinterval, while a Brownian increment over each fixed rational subinterval is a nondegenerate Gaussian and cannot be zero with positive probability.
Also $A(\infty)=\infty$ almost surely. If it were finite, the <finite-bracket convergence lemma> would make $H_t$ converge to a finite limit. Then $R_t^2=2t+O(1)$ on that event, forcing $\int_0^\infty R_t^2\,dt=\infty$, a contradiction. Thus no finite-lifetime extension is needed here.
Use the same inverse clock $T_u=\inf\{t:A(t)>u\}$ for both <martingales>, and set $W_u=H_{T_u}$, $B_u=Z_{T_u}$. Their bracket matrix is
$$
\begin{pmatrix}\langle W\rangle_u&\langle W,B\rangle_u\\\langle W,B\rangle_u&\langle B\rangle_u\end{pmatrix}
=\begin{pmatrix}u&0\\0&u\end{pmatrix}.
$$
The vector characterization proved in part (a) makes $(W,B)$ a two-dimensional <Brownian motion>; in particular its two coordinate processes are independent. Reversing the common clock gives
$$
\boxed{R_t^2=2W_{A(t)}+2t,\qquad Z_t=B_{A(t)},\qquad W\text{ and }B\text{ are independent}.}
$$
This is a <common-clock Brownian representation of radius and area>. Independence follows from the joint time change and identity bracket matrix; no independence of either <Brownian motion> from $A$ is asserted.
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