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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An example: $$ 1=\sqrt{2-1} \longrightarrow1=\sqrt{2-\sqrt{2-1}} $$ Repeat infinitely many times: $$ 1=\sqrt{2-\sqrt{2-\sqrt{2-\sqrt{2...}}}} $$
You can check this is true: $$ x=\sqrt{2-\sqrt{2-\sqrt{2-\sqrt{2...}}}} \longrightarrow x=\sqrt{2-x} $$ $$ x^2=2-x \longrightarrow x_1=1 $$ $x_2=-2$ can be discarded as we're taking a square root.
It's worth to note that what Prasun Biswas and I have posted here are expressions called "nested radicals".
Ramanujan was a mathematician that greatly raised interest and got some great conclusions about them at his time.
http://en.wikipedia.org/wiki/Nested_radical is worth a look.
Here's a nice "generator" of the result I commented in the comments.