Answer
The trinomial $x^{2}-2x+c$ is a perfect square when $c=1.$
Then $x^{2}-2x+1=(x-1)(x-1)=(x-1)^{2}$
Work Step by Step
$ x^{2}-2x+c\qquad$ ... find half the coefficient of $x$, which is $\displaystyle \frac{-2}{2}=-1$.
$\qquad$ ...square the result.
$(-1)^{2}=1\qquad$ ...substitute $c$ with $1$ in the original expression
$x^{2}-2x+1=(x-1)^{2}$
