Verify an implicit logarithm expansion by a second method

It is given that $y=\ln(2+\sin2x)$.

(i) Show that $e^y\dfrac{d^2y}{dx^2}+e^y\left(\dfrac{dy}{dx}\right)^2=-4\sin2x$.2 marks

(ii) By further differentiation of the above results, find the Maclaurin series for y, up to and including the term in $x^3$.3 marks

(iii) Verify that the series for $\ln(2+\sin2x)$ is the same as the result obtained in part (ii), if the standard series from the List of Formulae (MF27) are used.3 marks

(iv) Hence deduce the series expansion for $\dfrac{\ln(2+\sin2x)}{\sqrt{1-x}}$, up to and including the term in $x^2$.2 marks