√(1+√(1+√(1+√(1+...))) = 1+1/(1+1/(1+1/(1+...)))

Let the LHS = x

Then x^2 = 1 + x    or   x^2 - x -1 = 0 ........  (Eqtn 1)

Let the RHS = y.

Then y = 1 + 1/y   or     y^2 - y - 1 = 0  ........ (Eqtn 2)

Eqtns 1 and 2 are quadratics with the same solutions, so the LHS = RHS.

The solutions are phi, the Golden Ratio = (1 +- sqr(5) )/2.