(A) Given the equation $x+\frac{1}{x}=2$.Multiplying both sides by $x$,we get $x^2+1=2x$,which simplifies to $x^2-2x+1=0$.This is a perfect square: $(

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(A) Given the equation $x+\frac{1}{x}=2$.Multiplying both sides by $x$,we get $x^2+1=2x$,which simplifies to $x^2-2x+1=0$.This is a perfect square: $(x-1)^2=0$,which implies $x=1$.Now,substitute $x=1$ into the expression $x^{100}+\frac{1}{x^{100}}$:$1^{100}+\frac{1}{1^{100}} = 1+1 = 2$.Therefore,the value is $2$.