2022 NYJC Promo Q7

A curve has equation \(y=\sqrt{1+{{\tan }^{-1}}x}\). Show that \(y\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}+{{\left( \frac{\mathrm{d}y}{\mathrm{d}x} \right)}^{2}}=-\frac{x}{{{\left( 1+{{x}^{2}} \right)}^{2}}}\).

By further differentiation, obtain the Maclaurin series for \(y\), up to and including the term in \({{x}^{3}}\).

State the equation of the tangent to the curve at \(x=0\).[6]

Given that \(x\) is a sufficiently small angle, show that \(\sqrt{1+2\cos x}\approx \sqrt{3}\left( 1-\frac{{{x}^{2}}}{6} \right)\).

It is also given that \(1-10\sin x=\sqrt{1+2\cos x}\), form a quadratic equation in \(x\) and solve it.[4]