Question

xyz=1 then (1+x+y^-1) ^-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.