Factorise:

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

$=\left[x^{2}-2 \times x \times \frac{1}{x}+\left(\frac{1}{x}\right)^{2}\right]-y^{2}$

$=\left(x-\frac{1}{x}\right)^{2}-y^{2} \quad\left[a^{2}-2 a b+b^{2}=(a-b)^{2}\right]$

$=\left(x-\frac{1}{x}+y\right)\left(x-\frac{1}{x}-y\right) \quad\left[a^{2}-b^{2}=(a-b)(a+b)\right]$

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.