Question

If xyz=1 , then show that (1+x+y^{​​​​​​-1})^-1 + (1+y+z^{​​​​​​-1})^-1 + (1+z+x^{​​​​​​-1})^-1 =1

Answer 1

(1 + x + y^-1)^-1 +(1 + y + z^-1)^-1 + (1 + z + x^-1)^-1
= (1 + x + 1/y)^-1 +(1 + y + 1/z)^-1 + (1 + z + 1/x)^-1
= (1 + x + xz)^-1 +(1 + y + 1/z)^-1 + (1 + z + 1/x)^-1 [xyz = 1, or xz = 1/y]
= (1 + x + xz)^-1 +{(z + yz + 1)/z}^-1 + {(x + xz + 1)/x}^-1
= (1 + x + xz)^-1 +{(z + 1/x + 1)/z}^-1 + {(x + xz + 1)/x}^-1 [xyz = 1, or yz = 1/x]
= (1 + x + xz)^-1 +{(xz + 1 + x)/xz}^-1 + {(x + xz + 1)/x}^-1
= 1 / (1 + x + xz) + xz / (1 + x + xz) + x / (x + xz + 1)
= (1 + xz + x) / (1 + x + xz)
^{= 1}