Question

If xyz =1, then simplify:-
(1+x+y^-1)×(1+y+z^-1)^-1×(1+z+x^-1)^-1​

Answer 1

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.

:)