If you replace x2 − y2 with a, 2xy with b, and x2 + y2 with c, what is the identity equation?

X^2 - y^2 = a

2xy = b

x^2 + y^2 = c

A polynomial identity is an equation that is always true.`1

You can relate x^2 + y^2 with x^2 - y^2 and 2xy in this ways.

1) Step 1:

[x^2 + y^2]^2 = x^4 + 2(xy)^2 + y^4 = c^2

2) Step 2:

[x^2 - y^2] = x^4 - 2(xy)^2 + y^2 = a^2

 (2xy)^2 = 4(xy)^2 = b^2

x^4 - 2(xy)^2 + y^2 + (2xy)^2 = x^4 + 2(xy)^2 + y^2
        
            ↓                       ↓                        ↓
           a^2                    b^2    =              c^2

the identity is

c^2 = a^2 + b^2 equivalent to

(x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2

It will be a whole number.