This seems a little bit confusing because you commonly use x and y as variables making some reference to Cartesian plane. in particular, the formula x^2 + y^2 = 1 is the formula for a circle using Cartesian coordinates. That is not how x and y are being used here. They are cosine values associated with two angles. It would be easier to see if the author had called the variables a and b.
What is apparent here is that if the sum of the inverse cosines of the two measures is π/2, then the angles are complementary. Let us name the two angles a = Arccos x and b = Arccos y. Then, a = (90 - b), because they are complementary .
Now, sin a = cos (90-b) Which is a property of complementary angles
So, y = cos (90 - b) = sin a
so when the problem states x^2 + y^2 = 1
it is saying (cos a)^2 + (sin a)^2 = 1
which is the fundamental trigonometric identity.
