The following exercise is pulled from 'An introduction to Stochastic Differential Equations' by L.Evans.
Show that $X = (\cos(W),\sin(W))$ is the solution to the SDE $$\begin{align} dX^1 &= −\frac{1}{2}X^1dt − X^2dW, \\ dX^2 &= −\frac{1}{2}X^2dt + X^1dW. \end{align}$$
I have tried to use Ito's multi-dimensional formula, but without success. Any hints or references to relevant material would be appreciated.
The one-dimensional version of Ito's formula is enough here. If $Y$ is a continuous semi-martingale and $f$ is twice continuously differentiable, it yields
$$ df(Y_t)=f'(Y_t)dY_t+\frac12 f''(Y_t)d\langle Y\rangle_t.$$
Apply this with $Y=W$ and $f=\cos$ or $f=\sin$. Recall that $\langle W\rangle_t=t$. This way you can show that $(\cos(W),\sin(W))$ is a solution to this SDE. In order to prove uniqueness (for the initial value $(X_0^1,X_0^2)=(\cos(W_0),\sin(W_0))=(1,0)$), note that the coefficients of this SDE are Lipschitz and recall Ito's uniqueness theorem.
More generally, you can check in the same way that the unique strong solution for given initial values $(X_0^1,X_0^2)\in\Bbb R^2$ is
$$ \begin{pmatrix} X^1 \\ X^2 \end{pmatrix}=\begin{pmatrix} -X^2_0 \sin(W) + X^1_0 \cos(W) \\ X^1_0 \sin(W) + X^2_0 \cos(W) \end{pmatrix}.$$