2025 SAJC Promo Q9

It is given that \(y=\ln(\cos 2x+1),\)

1. Show that \(\frac{\mathrm{d}^2y}{\mathrm{d}x^2}+\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2+4(1-e^{-y})=0.\)

   [3]
2. Hence, obtain the Maclaurin expansion of \(y\) in terms of \(x\) up to and including the term in \(x^2\).[3]
3. Using standard series from the List of Formulae (MF27), expand \(\ln(\cos 2x+1)\) as far as the term in \(x^3\) and verify the result obtained in part (b).[4]