If xyz=1 then simplify (1+x+y-1)-1 . . .

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

Doubt by Balvinder 

Solution : 
xyz=1
xy=1/z
1/z=xy— (1) 

Also 
z=1/xy — (2) 

Now, 

(1+x+y^-1)^-1
=[1+x+(1/y)]^-1
=[(y+xy+1)/y]^-1
=[y/(y+xy+1)]
=[y/(1+y+xy)] — (3)

(1+y+z^-1)^-1
=[1+y+(1/z)]^-1
=[1+y+xy]^-1 [Using equation (1)]
=[1/(1+y+xy)] — (4) 

(1+z+x^-1)^-1
=[1+z+(1/x)]^-1
=[1+(1/xy)+(1/x)]^-1 [Using equation (2)]
=[(xy+1+y)/xy]^-1
=[xy/(xy+1+y)]
=[xy/(1+y+xy)] — (5)

Adding equation (3), (4) and (5) 

(1+x+y^-1)^-1×(1+y+z^-1)^-1×(1+z+x^-1)^-1
= [y/(1+y+xy)] + [1/(1+y+xy)] + [xy/(1+y+xy)] 
=[(y+1+xy)/(1+y+xy)]
=[(1+y+xy)/(1+y+xy)]
= 1 

Hence
(1+x+y^-1)^-1×(1+y+z^-1)^-1×(1+z+x^-1)^-1 = 1