 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2023/paper-1/15/c/solution

Factor the leading term out of the Maclaurin series for the sine:

sin x = x (1 - \frac{x ^{2}}{6} + \frac{x ^{4}}{120} + O (x^{6})) .

The binomial theorem gives, through degree four inside the bracket,

(1 - \frac{x ^{2}}{6} + \frac{x ^{4}}{120} + O (x^{6}))^{6} = 1 + 6 (- \frac{x ^{2}}{6} + \frac{x ^{4}}{120}) + (2 6) \frac{x ^{4}}{36} + O (x^{6}) .

x^{4}

1/20 + 5/12 = 7/15

. Multiplying back by

x^{6}

yields

sin^{6} x = x^{6} - x^{8} + \frac{7}{15} x^{10} + O (x^{12}) .

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