Question

Prove that $xy=\frac{1}{4}\left[\right(x+y{)}^2-(x-y{)}^2]$.

Answer 1

$xy=\frac{1}{4}\left[\right(x+y{)}^2-(x-y{)}^2]$
First take R.H.S
$\frac{1}{4}\left[\right(x+y{)}^2-(x-y{)}^2]$
We know that $(x+y{)}^2=x^2+2xy+y^2,(x-y{)}^2=x^2-2xy+y^2$
Therefore, $\frac{1}{4}[x^2+2xy+y^2-x^2+2xy-y^2]$
$=$ $\frac{1}{4}[2xy+2xy]$
$=$ $\frac{1}{4}\left[4xy\right)]$ (cancelling 4)
$=$ $xy$
So, L.H.S $=$ R.H.S
$xy=xy$
Hence $xy=\frac{1}{4}\left[\right(x+y{)}^2-(x-y{)}^2]$ is proved.