xyz=1 then (1+x+y^-1) ^-1*(1+y+z^-1)^-1*(1+z+x^-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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