Zeroes of $dx_1$ on $\mathbb{R}^2$ vs. zeroes of $dx_1|_{S^1}$ on $S^1$

Question

Let us consider $S^1$ as a manifold embedded in $\mathbb{R}^2$. Let $dx_1\in\Omega^1(\mathbb{R^2})$. $$ Z_{\mathbb{R}^2}:=\{p\in\mathbb{R}^2:(dx_1)_p=0\}\\ Z_{S^1}:=\{p\in\mathbb{R}^2:(dx_1|_{S^1})_p=0\} $$ I have to prove that $Z_{\mathbb{R}^2}\cap S^1\ne Z_{S^1}$.

I have checked that $Z_{\mathbb{R}^2}\cap S^1\subset Z_{S^1}$. Now I have to prove the inequality. Any help?