About the algebra $H^\infty + C(\mathbb{T})$

Question

We know that $H^\infty + C(\mathbb{T})$ is the closed subalgebra of $L^\infty(\mathbb{T})$ containing $H^\infty$. How to show that $H^\infty + C(\mathbb{T})$ = clos$[\cup_{n\geq 0} \chi_{-n} H^\infty$], where $\chi_n (e^{it})=e^{int}$ for $n\in \mathbb{Z}$ ? Also we have $H^\infty + C(\mathbb{T})$ = clos$[H^\infty + \mathscr{P}]$ where $\mathscr{P}$ is the collection of all trigonometric polynomials in $\mathbb{T}$. Then How to show that $H^\infty + \mathscr{P}$ = $\{\psi\chi_{-n} : \psi\in H^\infty, n\geq 0\}$ ?