= Solution
To obtain a <basin estimate from a Lyapunov sublevel set>, take the largest sublevel component around the origin that remains inside the strict-decrease rectangle. On its boundary $|x|=\sqrt2$ or $|y|=\pi/2$, the minimum value of $V$ is $1$. Hence the maximal open estimate is
$$
\boxed{
\mathcal D_V=
\left\{(x,y): |y|<\frac\pi2,
\ \frac{x^2}{2}+\sin^2y<1\right\}}.
$$
Every smaller closed sublevel set $V\leq c<1$ is compact and positively invariant, and $\dot V<0$ there except at the origin. The <LaSalle invariance principle> therefore implies convergence to the origin. Taking the union over $c<1$ shows that $\mathcal D_V$ lies in the domain of stability. No larger sublevel component can be certified from this $V$ alone, because every level $c>1$ contains points $(0,y)$ just beyond $|y|=\pi/2$ where $\dot V>0$.
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