Question

question no .5
please help

Answer 1

given xyz=1
xy = 1/z, yz = 1/x , xz = 1/y
now
=[ 1+ x + 1/ y]-1+ [ 1 + y + 1 / z ]-1+ [ 1 + z + 1 /x ]-1
= 1 / ( 1+ x + 1/ y) + 1 / (1 + y + 1 / z) + 1 /(1 + z + 1 /x)
now consider
1/ (1 + x + 1/y) = y / (y + xy + 1)
= y / (y + 1/z + 1) = yz / (yz + 1 + z)
= yz / ( 1/x + 1 + z) = (1/x)/ ( 1/x + 1 + z)
From above we get [using ratio and proportion a/c= b/d = (a+b)/(c+d)]
1/ (1 + x + 1/y) = y / (y + 1/z + 1)
= (1/x) / (1/x + 1 + z)
= [1+ y + 1/x] / [(1 + x + 1/y) + (y + 1/z + 1) + ( 1/x + 1 + z)] --------------------(A)
Or
1/ (1 + x + 1/y) = [1+ y + 1/x] / [(1 + x + 1/y) + (y + 1/z + 1) + ( 1/x + 1 + z)] -------(1)
Replacing x by y and y by z and z by x we get
1/ (1 + y + 1/z) = [1+ z + 1/y] / [(1 + y + 1/z) + (z + 1/x + 1) + (1/y + 1 + x)] -------(2)
And again x by y and y by z and z by x we get
1/ (1 + z + 1/x) = [1+ x + 1/z] / [(1 + z + 1/x) + (x + 1/y + 1) + (1/z + 1 + y)] --------- ( 3)
We observe that in all cases 1, 2 , 3 the denominator is same
or
= [(1+1+1) + (x + y+ z) + (1/x + 1/y +1/z)]
Now adding LHS of (1 + 2 + 3) = Adding RHS OF (1 + 2 + 3)
1/ (1 + x + 1/y) + 1/ (1 + y + 1/z) + 1/ (1 + z+ 1/x)
= {[1+ y + 1/x] +[1+ z + 1/y] + [1+ x + 1/z]} /[(1+1+1) + (x + y + z) + (1/x + 1/y +1/z)]
= 1

Answer 2

hope it helps...... :)