The Golden Equation

Algebra Level 4

\[\sqrt{1+\frac{1}{1+x}}+\frac{1}{\sqrt{1+x}} = \sqrt{x}+\frac{1}{\sqrt{x}}\]

Which of the following is the root of the above equation?

Note: \(\phi=\frac{\sqrt5+1}{2}\) and denotes the golden ratio.