For the first identity start with (x+y+z)^3 = 0

Expand the left side:

x^3 + y^3 + z^3 + 3x^2y + 3xy^2 + 3x^2z + 3xz^2 + 3y^2z + 3yz^2 + 6xyz = 0

Now just subtract x^3 + y^3 + z^3 from each side and divide by xyz:

3(x/y + y/x + x/z + z/x + y/z + z/y) + 6 = -(x^2/(yz) + y^2/(xz) + z^2/(xy))

This is our first identity.

For the second identity start with the less obvious (-x+y+z)*(x-y+z)*(x+y-z) = (x+y+z - 2x)*(x+y+z - 2y)*(x+y+z - 2z) = -8xyz.

Divide the left side and right side by xyz.  On the left, the first term gets divided by x, the second by y and the third by z:

(-1 + y/x + z/x)*(-1 + x/y + z/y)*(-1 + x/z + y/z) = -8

Now expand the product and simplify like terms:

(x/y + y/x + x/z + z/x + y/z + z/y) - (x^2/(yz) + y^2/(xz) + z^2/(xy)) = -6

This is our second identity.

Now substitute the first identity into the second identity:

(x/y + y/x + x/z + z/x + y/z + z/y) + 3(x/y + y/x + x/z + z/x + y/z + z/y) + 6 = -6

Now this simplifies into a very nice symmetric identity:

At this point we can finally introduce the second constraint x/y + y/z + z/x = x/z + z/y + y/x + 1.  Just by adding one to each side of the symmetric identity, the two sides of the problem's identity add up to -2, but since they are also equal to each other, they must each equal -1.  That is the answer: x/y + y/z + z/x = x/z + z/y + y/x + 1 = -1.