Recover a logarithmic series from a differential identity

Let $y=\ln(1+\sin x)$.

(i) Show that $e^y\dfrac{d^2y}{dx^2}=1-e^y[(\dfrac{dy}{dx})^2+1]$.

(ii) Find its Maclaurin series through $x^2$.

(iii) Verify it using the standard series for $\sin x$ and $\ln(1+x)$.