In reply to Denis Jaisson.

No, I don't know the name of this equation. It is a tautology, because
sqrt(1+x) = 1 + x/(1 + sqrt(1+x))
I think you can derive it by using the continued fraction algorithm. From this equation, you can recursively replace the sqrt(1+x) term on the right side to go to any depth.
You might want to study the continued fraction algorithm by starting with the continued fraction for sqrt(2). See https://www.cut-the-knot.org/proofs/SqContinuedFraction.shtml.
See if you can replace '2' with '1+x' and perform the same steps.