The Golden Equation

Algebra Level 3

$\sqrt{1+\frac{1}{1+x}}+\frac{1}{\sqrt{1+x}} = \sqrt{x}+\frac{1}{\sqrt{x}}$

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

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

\phi-2

\phi-1

\dfrac{\phi}{3}

\dfrac{\phi}{2}

\phi^2

None of the above