Question Number 43707 by Tawa1 last updated on 14/Sep/18
$$\mathrm{Simplify}:\:\:\: \\ $$$$\left(\mathrm{x}\:+\:\mathrm{y}\:+\:\mathrm{z}\right)\left(\mathrm{x}^{−\mathrm{1}} \:+\:\mathrm{y}^{−\mathrm{1}} \:+\:\mathrm{z}^{−\mathrm{1}} \right)\:=\:\left(\mathrm{x}^{−\mathrm{1}} \:\mathrm{y}^{−\mathrm{1}} \:\mathrm{z}^{−\mathrm{1}} \right)\left(\mathrm{x}\:+\:\mathrm{y}\right)\left(\mathrm{y}\:+\:\mathrm{z}\right)\left(\mathrm{z}\:+\:\mathrm{x}\right) \\ $$
Commented by math1967 last updated on 14/Sep/18
$${Simplify}\:???/{prove}?? \\ $$
Commented by Tawa1 last updated on 14/Sep/18
$$\mathrm{It}\:\mathrm{should}\:\mathrm{be}\:\mathrm{prove}\:\mathrm{sir} \\ $$
Commented by Tawa1 last updated on 14/Sep/18
$$\mathrm{thanks} \\ $$
Answered by math1967 last updated on 14/Sep/18
$${I}\:{think}\:{L}.{H}.{S} \\ $$$$=\left({x}+{y}+{z}\right)\left({x}^{−\mathrm{1}} +{y}^{−\mathrm{1}} +{z}^{−\mathrm{1}} \right)−\mathrm{1} \\ $$$$=\left({x}+{y}+{z}\right)\left(\frac{{yz}+{zx}+{xy}}{{xyz}}\right)−\mathrm{1} \\ $$$$=\frac{\left({x}+{y}+{z}\right)\left({xy}+{yz}+{zx}\right)−{xyz}}{{xyz}} \\ $$$$=\frac{{z}\left({x}+{y}\right)^{\mathrm{2}} +{xy}\left({x}+{y}\right)+{z}^{\mathrm{2}} \left({x}+{y}\right)+{xyz}−{xyz}}{{xyz}} \\ $$$$=\frac{\left({x}+{y}\right)\left({zx}+{zy}+{xy}+{z}^{\mathrm{2}} \right)}{{xyz}} \\ $$$$=\frac{\left({x}+{y}\right)\left({y}+{z}\right)\left({z}+{x}\right)}{{xyz}} \\ $$$$=\left(\frac{\mathrm{1}}{{xyz}}\right)\left({x}+{y}\right)\left({y}+{z}\right)\left({z}+{x}\right) \\ $$$$=\left({x}^{−\mathrm{1}} {y}^{−\mathrm{1}} {z}^{−\mathrm{1}} \right)\left({x}+{y}\right)\left({y}+{z}\right)\left({z}+{x}\right) \\ $$$$={R}.{H}.{S} \\ $$
Commented by Tawa1 last updated on 14/Sep/18
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}\: \\ $$