If $$a+\frac{1}{a-2}=4$$, then the value of $$(a-2)^{2}+(\frac{1}{a-2})^{2}$$ is
$$a+\frac{1}{a-2}=4$$
subtract 2 on both sides,
$$(a-2)+\frac{1}{a-2}=4-2$$
$$(a-2)+\frac{1}{a-2}=2$$
squaring on both sides
$$[(a-2)+\frac{1}{a-2}]^{2}=4$$
$$(a-2)^{2}+(\frac{1}{a-2})^{2}+2(a-2).(\frac{1}{a-2})=4$$
$$(a-2)^{2}+(\frac{1}{a-2})^{2}+2=4$$
$$(a-2)^{2}+(\frac{1}{a-2})^{2}=4-2=2$$
so the answer is option B.
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