Density of $C_c^{\infty}(\mathbb{R}^2)$ in Sobolev space $H^{\frac{1}{2}}$

Question

Define space $E=\{u\in H^{\frac{1}{2}}(\mathbb{R}^2) : \frac{\partial u}{\partial y}\in L^2(\mathbb{R}^2)\}$, its norm defined by $||u||_E^2=||u||_{H^{\frac{1}{2}}}^2+||\frac{\partial u}{\partial y}||_{L^2}^2$.

Are $C_c^{\infty}(\mathbb{R}^2)$ functions dense in this space?

Moreover, how to show: $\forall u\in E$, we have $u(x,0)\in L^2(\mathbb{R}^2)$.

Any suggestions are welcome! Thanks!