Question

Expand the following, using suitable identity.
${(\frac{x}{y}+\frac{y}{z}+\frac{z}{x})}^2$

Answer 1

We have, ${(\frac{x}{y}+\frac{y}{z}+\frac{z}{x})}^2$

$(a+b+c{)}^2=a^2+b^2+c^2+2ab+2bc+2ca$
$={\left(\frac{x}{y}\right)}^2+{\left(\frac{y}{z}\right)}^2+{\left(\frac{z}{x}\right)}^2+2.\frac{x}{y}.\frac{y}{z}+2.\frac{y}{z}.\frac{z}{x}+2.\frac{z}{x}.\frac{x}{y}$
$=\frac{x^2}{y^2}+\frac{y^2}{z^2}+\frac{z^2}{x^2}+2\frac{x}{z}+2\frac{y}{x}+2\frac{z}{y}$