Prove $D(A):= {win H^{2}(0, 2pi): w(0) = w(2pi), w^{'}(0) = w^{'}(2pi) }$ is dense in $L^{2}(0,2pi).$

Let $H^{2}(0, 2pi)$ and $L^{2}(0, 2pi)$ be standard notation for the well-known functional spaces.

Prove

$$D(A):= {win H^{2}(0, 2pi): w(0) = w(2pi), w^{‘}(0) = w^{‘}(2pi) }; is; dense; in; L^{2}(0,2pi).$$

What I was trying is to prove that $C_{c}^{infty}([0, 2pi])$ is dense in $L^{2}(0, 2pi).$

Thanks in advance.