Data Sufficiency (DS) | Re: If y≠1, is x=1?

Got this question in GMATPrep EP1 and below is the approach I followed:
1. x^2 + y^2 = 1
Not given that x and y are integers. Hence, two cases possible:
a. If y = 0, x = 1 or x = -1
b. If y =1/sqrt2 , x = 1/sqrt2. Not sufficient

2. y = 1-x
x can take any value except for zero. Hence not sufficient.

(1) + (2),
x^2 + y^2 = 1 and y = 1-x
We know that (x+y)^2 = x^2 + y^2 + 2xy
1^2 = 1 + 2xy
2xy = 0
Given that y ≠ 1, for xy to be zero x has to be 1. Sufficient

Hence option C.

Kudos if it helps
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