2025 TJC Promo Q10

It is given that \(y=\sqrt{1+\sin x}\).

1. Show that \(2y\frac{d^2y}{dx^2}+2\left(\frac{dy}{dx}\right)^2+y^2=1\).[3]
2. By further differentiation of the result in (a), find the first 4 terms of the Maclaurin expansion of \(\sqrt{1+\sin x}\).[4]
3. By using appropriate standard series from the List of Formulae (MF27), verify that the expansion found in (b) is correct.[3]