My first reaction was to evaluate this expression using a calculator, to see if that gave any clues. The answer came out to be exactly(?) 1. So if the answer really is as tidy as that, there should be some way to find it without too much work.

Let and . Then , which factorises as . We want to find , so let . Then

Also, . Hence , so that , and equation (*) becomes , or . The only real solution to that cubic equation is z=1, which is what we wanted.

Taking this a bit further, the fact that x and y satisfy the equations and imply that , which is the equation for the golden ratio . Thus , as you can verify by checking that .