Factorise completely $x^4 - y^4$. Answer ... Given that $x^2 - 2xy = 1 - y^2$, express $y$ in terms of $x$. Answer ... or ...

1. Apply the difference of two squares identity to the expression

$x^4 - y^4 = (x^2)^2 - (y^2)^2$
$= (x^2 - y^2)(x^2 + y^2)$

2. Factorise completely by applying the difference of two squares again

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

1. Rearrange the equation to group all terms on one side or form a perfect square.

$y^2 - 2xy + x^2 = 1$

2. Factorise the left-hand side as a perfect square.

$(y - x)^2 = 1$

3. Solve for $y$ by taking the square root of both sides.

$y - x = \pm 1$
$y = x + 1 \quad \text{or} \quad y = x - 1$