2017 HCI P1 Q7

It is given that \(\ln y=2\sin x\). Show that \(\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}=-y\ln y+\frac{1}{y}{{\left( \frac{\mathrm{d}y}{\mathrm{d}x} \right)}^{2}}\).[2]

Find the first four terms of the Maclaurin series for \(y\) in ascending powers of \(x\).[4]

Using appropriate expansions from the List of Formulae (MF26), verify the expansion found in part (ii).[2]

Given that \(x\) is sufficiently small for \({{x}^{4}}\) and higher powers of \(x\) to be neglected, deduce an approximation for \({{\mathrm{e}}^{\left( 2\sin x \right)-\ln \left( \sec x \right)}}\) in ascending powers of \(x\).[2]