\(\frac{x+2+\sqrt{x^2-4}}{x+2-\sqrt{x^2-4}}+\frac{x+2-\sqrt{x^2-4}}{x+2+\sqrt{x^2-4}}\)
\(=\frac{\left(\sqrt{x+2}\right)^2+\sqrt{x-2}\cdot\sqrt{x+2}}{\left(\sqrt{x+2}\right)^2-\sqrt{x-2}\cdot\sqrt{x+2}}+\frac{\left(\sqrt{x+2}\right)^2-\sqrt{x-2}\cdot\sqrt{x+2}}{\left(\sqrt{x+2}\right)^2+\sqrt{x-2}\cdot\sqrt{x+2}}\)
\(=\frac{\sqrt{x+2}\left(\sqrt{x+2}+\sqrt{x-2}\right)}{\sqrt{x+2}\left(\sqrt{x+2}-\sqrt{x-2}\right)}+\frac{\sqrt{x+2}\left(\sqrt{x+2}-\sqrt{x-2}\right)}{\sqrt{x+2}\left(\sqrt{x+2}+\sqrt{x-2}\right)}\)
\(=\frac{\sqrt{x+2}+\sqrt{x-2}}{\sqrt{x+2}-\sqrt{x-2}}+\frac{\sqrt{x+2}-\sqrt{x-2}}{\sqrt{x+2}+\sqrt{x-2}}\)
\(=\frac{\left(\sqrt{x+2}+\sqrt{x-2}\right)^2+\left(\sqrt{x+2}-\sqrt{x-2}\right)^2}{\left(\sqrt{x+2}-\sqrt{x-2}\right)\left(\sqrt{x+2}+\sqrt{x-2}\right)}\)
\(=\frac{x+2+x-2+2\sqrt{\left(x+2\right)\left(x-2\right)}+x+2+x-2-2\sqrt{\left(x+2\right)\left(x-2\right)}}{x+2-x+2}\)
\(=\frac{4x}{4}\)
\(=x\)
