Question

Factorise:
$x^2-2+\frac{1}{x^2}{y}^2$

Answer 1

$$\begin{gathered} x^2-2+\frac{1}{x^2}-y^2 \\ =\left[x^2-2\times x\times \frac{1}{x}+{\left(\frac{1}{x}\right)}^2\right]-y^2 \\ ={\left(x-\frac{1}{x}\right)}^2-y^2\left[a^2-2ab+b^2={\left(a-b\right)}^2\right] \\ =\left(x-\frac{1}{x}+y\right)\left(x-\frac{1}{x}-y\right)\left[a^2-b^2=\left(a-b\right)\left(a+b\right)\right] \end{gathered}$$

Disclaimer: The expression of the question should be $x^2-2+\frac{1}{x^2}-y^2$. The same has been done before solving the question.