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$\left ( \frac{1}{\sqrt{p}+\sqrt{p+1}}+\frac{1}{\sqrt{p}-\sqrt{p+1}}\right ): \left (1+\sqrt{\frac{p+1}{p-1}} \right )=\\\\=\frac{\sqrt{p}-\sqrt{p+1}+\sqrt{p}+\sqrt{p+1}}{(\sqrt{p}+\sqrt{p+1})(\sqrt{p}-\sqrt{p+1})}:\frac{\sqrt{p-1}+\sqrt{p+1}}{\sqrt{p-1}}=\\\\= \frac{2\sqrt{p}}{p-(p+1)}\cdot \frac{\sqrt{p-1}}{\sqrt{p-1}+\sqrt{p+1}} = \frac{2p}{-1}\cdot \frac{\sqrt{p-1}}{\sqrt{p-1}+\sqrt{p+1}} =- \frac{2p\sqrt{p-1}}{\sqrt{p-1}+\sqrt{p+1}}$