2024 ACJC Promo Q2

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

1. Show that \(2y\frac{\mathrm{d}y}{\mathrm{d}x}=1+{{\left( {{y}^{2}}-1 \right)}^{2}}\) and \(y\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}+{{\left( \frac{\mathrm{d}y}{\mathrm{d}x} \right)}^{2}}=2y\left( {{y}^{2}}-1 \right)\left( \frac{\mathrm{d}y}{\mathrm{d}x} \right)\).[3]
2. Hence find the first three terms in the Maclaurin expansion of \(y\).[2]