if xyz=1 then show that (1+x+y^-1)^-1 + (1+y+z^-1)^-1 + (1+z+x^-1)^-1 = 1
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Step-by-step explanation:
Given:— xyz = 1
To prove:- (1+x+y^−1)^−1+(1+y+z^−1)^−1+(1+z+x^−1)^−1 = 1
Proof:-
Taking L.H.S.-
(1+x+y^−1)^−1+(1+y+z^−1)^−1+(1+z+x^−1)^−1
= (1+x+xz)^−1+(1+y+xy)^−1+(1+z+yz)^−1
= (1+x+xz)^−1+(xyz+y+xy)^−1+(1+z+yz)^−1
= (1+x+xz)^ −1+y^−1(1+x+xz)^−1+(1+z+yz)^−1
= (1+x+xz)^−1(1+y^−1)+(1+z+yz)^−1
= (xyz+x+xz)^−1(1+y^−1)+(1+z+yz)^−1
= x^−1(1+z+yz)^−1(1+y^−1)+(1+z+yz)^−1
= (1+z+yz)^−1(x^−1+(xy)^−1)+(1+z+yz)^−1
= (1+z+yz)^−1(yz+z+1)
= 1+z+yz/1+z+yz
= 1
= R.H.S.
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