1 1 1 If xyz = 1 (x, y, * 0), then xy 1+y+z²³1+2+x²² 1+x+y equals
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Answer:
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
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