Question

Доведіть тотожність!!!

Answer 1

$\displaystyle \frac{3xyz}{xy + yz + zx } + \cfrac{\cfrac{1-x}{x} + \cfrac{1-y}{y}+ \cfrac{1-z}{z} }{\cfrac{1}{x} +\cfrac{1}{y} +\cfrac{1}{z} } = \\\\\\ = \frac{3xyz}{xy + yz + zx } + \cfrac{\cfrac{(1-x)yz + (1-y)xz + (1-z)xy}{xyz} }{\cfrac{xy + yz + xz}{xyz} } = \\\\\\ =\frac{3xyz}{xy + yz + zx} + \frac{yz - xyz + xz - xyz + xy - xyz}{xy + yz + zx} = \\\\\\ =\frac{3xyz }{xy+ yz + zx} + \frac{xy + yz + xz - 3xyz}{xy+ yz + zx} = \frac{xy + yz + xz+ 3xyz -3xyz}{xy+ yz + zx} =$

$=\dfrac{xy + yz + xz}{xy+ yz + zx} = 1$

Доказано !