Let $A$ be a unital algebra over $\mathbb{C}$, $C^n(A)$ be a space off all $n+1$-linear maps $f:A^{n+1} \to \mathbb{C}$ (to be called $n$ cochains) and define $b:C^n(A) \to C^{n+1}(A)$ by the formula $$(bf)(a_0,a_1,...,a_n,a_{n+1}:=\sum_{j=0}^n(-1)^jf(a_0,...,a_{j-1},a_ja_{j+1},a_{j+2},...,a_{n+1}+(-1)^{n+1}f(a_{n+1}a_0,a_1,...,a_n).$$ The cohomology of this complex is called Hochschild cohomology. Let us call the $n$-cochain normalized if $f(a_0,a_1,...a_{n+1})=0$ whether $a_i=1$ for some $i \geq 1$. One checks that the normalized cochains form a subcomplex of $(C^*(A),b)$

How to prove that the inclusion of normalized cochains into all cochains induces an isomorphism in cohomology?

Let $A$ be a unital algebra over $\mathbb{C}$. Let $C^n(A)$ be the space of all $n+1$-linear maps $f:A^{n+1} \to \mathbb{C}$ (to be called $n$-cochains). Define $b:C^n(A) \to C^{n+1}(A)$ by the formula $$(bf)(a_0,a_1,...,a_n,a_{n+1}) \\ :=\sum_{j=0}^n(-1)^jf(a_0,...,a_{j-1},a_ja_{j+1},a_{j+2},...,a_{n+1}+(-1)^{n+1}f(a_{n+1}a_0,a_1,...,a_n).$$ The cohomology of this complex is called Hochschild cohomology of $A$. Let us call the $n$-cochain $f$ normalized if $f(a_0,a_1,...,a_n)=0$ whether $a_i=1$ for some $i \geq 1$. One checks that the normalized cochains form a subcomplex of $(C^*(A),b)$.

How to prove that the inclusion of normalized cochains into all cochains induces an isomorphism in cohomology?