$[W_t^1,...,W_t^n]$ - n-dimensional Wiener process, $R_t=\left(\sum_{n=1}^n(W_t^i)^2\right)^{\frac{1}{2}}, M_t=\sum_{i=1}^{n} \int_0^t\frac{W_s^i}{R_s}dW_s^i$ and I have to show that
$dR_t=dM_t+\frac{n-1}{2R_t}dt$
I try to use multidimensional Ito Formula to solve it but I didn't get anything sensible.
Suggestion: It may be easier to proceed in two steps. First apply Ito to $S_t:=R_t^2 =f(W_t)$, where $f(x):=\|x\|^2$ (so that $\nabla f(x) =2x$ and $\Delta f(x) = 2n$). This should yield $$ dS_t = W_t\cdot dW_t+ n\,dt= R_t\,dM_t+n\,dt. $$ (Notice that that $\langle S\rangle_t = 4\int_0^t W_s^2\,ds$.) Then apply Ito a second time to $R_t=S^{1/2}_t$.