So complicated

Algebra Level 3

Given \(x\) and \(y\) are distinct real numbers such that \[\frac{x^{2}+y^{2}}{x^{2}-y^{2}}+\frac{x^{2}-y^{2}}{x^{2}+y^{2}}=4\] If \[\frac{x^{8}+y^{8}}{x^{8}-y^{8}}+\frac{x^{8}-y^{8}}{x^{8}+y^{8}}\] can be expressed as \(\frac{a}{b}\), where \(a\) and \(b\) are positive numbers and \(\text{gcd}(a,b)=1\), find \(a+b\).

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