Statement (A): ${\left(\frac{x^a}{x^b}\right)}^c{\left(\frac{x^b}{x^c}\right)}^a{\left(\frac{x^c}{x^a}\right)}^b=1$

Statement (B): $(x+y){(x^{-1}+y^{-1})}^{-1}=xy$

$\text{A:\hspace{0.17em}\hspace{0.17em}}{\left(\frac{x^a}{x^b}\right)}^c{\left(\frac{x^b}{x^c}\right)}^a{\left(\frac{x^c}{x^a}\right)}^b$

$={\left(x^{a-b}\right)}^c.{\left(x^{b-c}\right)}^a.{\left(x^{c-a}\right)}^b$

$=x^{(a-b)c}.x^{(b-c)a}.x^{(c-a)b}$

$=x^{ac-bc}.x^{ba-ca}.x^{cb-ab}$

$x^{ac-bc+ab-ac+bc-ab}=x^0=1$

$\text{B:\hspace{0.17em}}(x+y){(x^{-1}+y^{-1})}^{-1}$

$=(x+y){(\frac{1}{x}+\frac{1}{y})}^{-1}$

$=(x+y)\left(\frac{xy}{(x+y)}\right)=xy$