Question

Simplify: ${(x+\frac{1}{x})}^2-{(x-\frac{1}{x})}^2$

Answer 1

Formula:

$a^2-b^2=(a+b)(a-b)$

$(x+\frac{1}{x}{)}^2-(x-\frac{1}{x}{)}^2$ is of the form $a^2-b^2$

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

$\Rightarrow (x+\frac{1}{x}+x-\frac{1}{x})(x+\frac{1}{x}-x+\frac{1}{x})$

$\Rightarrow (x+x)(\frac{1}{x}+\frac{1}{x})$

$\Rightarrow \left(2x\right)\left(\frac{2}{x}\right)$(Cancel $x$)

$\Rightarrow 2\times 2=4$