 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2004/paper-2/8/a/ii/solution

(1 + u)^{- 1} = 1 - u + u^{2} + O (u^{3})

with

u = sin x = x + O (x^{3})

. The constant term counts as the first nonzero term, so the requested expansion is

\frac{1}{1 + sin x} = 1 - x + x^{2} + O (x^{3}) .

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