Question:
Factorise:
$x^{2}-2+\frac{1}{x^{2}} y^{2}$
Solution:
$x^{2}-2+\frac{1}{x^{2}}-y^{2}$
$=\left[x^{2}-2 \times x \times \frac{1}{x}+\left(\frac{1}{x}\right)^{2}\right]-y^{2}$
$=\left(x-\frac{1}{x}\right)^{2}-y^{2} \quad\left[a^{2}-2 a b+b^{2}=(a-b)^{2}\right]$
$=\left(x-\frac{1}{x}+y\right)\left(x-\frac{1}{x}-y\right) \quad\left[a^{2}-b^{2}=(a-b)(a+b)\right]$
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.
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