From distance preservation to inner-product preservation

Assume a linear map $A$ satisfies $(1-\eps)\|v\|_2^2 \le \|Av\|_2^2 \le (1+\eps)\|v\|_2^2$ for all $v$ in the set $\{x_i\pm x_j\}$. Prove that for all $i,j$,

$$\big|\langle Ax_i,Ax_j\rangle - \langle x_i,x_j\rangle\big| \le \frac\eps2\big(\|x_i\|_2^2+\|x_j\|_2^2\big).$$

Deduce that unit vectors have their cosine similarities preserved to additive $\eps$.