How to generate complicated looking identities such as $\sqrt [3] {2 + \sqrt 5} - \sqrt [3] {2 - \sqrt 5}=1$ easily? How to generate complicated looking identities, or even more complicated looking identies such as $\sqrt [3] {2 + \sqrt 5} - \sqrt [3] {2 - \sqrt 5}=1$ easily? I saw the identity to be shown. What is I think would be more interesting would be the origin of what ever it is that can produce even more complicated looking identities with relative ease (by relative ease I mean subbing a value and generating the identity in a mechanical fashion, no matter how laborious or long that procedure takes)

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Here's a nice "generator" of the result I commented in the comments.