Recover an inverse-cosine tangent from its Maclaurin coefficients

Given $\cos y=\sqrt{1-e^x/4}$ and $0<y<\pi/2$, show $\sin(2y)y^{\prime}=e^x/4$. By further differentiation find the Maclaurin series for $y$ through $x^2$ in exact form.5 marks

Deduce the tangent to $y=\cos^{-1}\!\sqrt{1-e^x/4}$ at $x=0$.2 marks