2024 EJC P1 Q7

For a right-angled triangle \(ABC\) with \(BC=a\) and \(AC=b\), \(\measuredangle CAB=x\) and \(\measuredangle ABC=\frac{\pi }{2}\) as shown below, a sector with radius \(a\) is extended from the vertex \(C\). This sector meets the side \(AC\) at the point \(Y\).

1. Show that \({{\left( \frac{AC}{AY} \right)}^{2}}=\frac{1}{{{\left( 1-\sin x \right)}^{2}}}\).[2]
2. Using appropriate expansions from the List of Formulae (MF26), find the Maclaurin series for \(\mathrm{f}(x)={{\left( 1-\sin x \right)}^{-2}}\), up to and including the term in \({{x}^{3}}\).[3]
3. Deduce the series expansion for \(\frac{2\cos x}{{{\left( 1-\sin x \right)}^{3}}}\), up to and including the term in \({{x}^{2}}\).[2]
4. Using your answer in part (b) or otherwise, show that \(\frac{AC}{AY}\approx 1+x+{{x}^{2}}\), when \(x\) is sufficiently small.[3]