 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2010/paper-1/20/c/i/solution

The cosine-difference identity gives

cos x - cos 3 x = 2 sin 2 x sin x

. Hence

\frac{c o s x - c o s 3 x}{x ^{2}} = 4 \frac{s i n 2 x}{2 x} \frac{s i n x}{x} .

Each sine ratio tends to one, so

x \to 0 lim \frac{cos x - cos 3 x}{x ^{2}} = 4.

Equivalently, the quadratic terms of the two cosine expansions differ by

4 x^{2}

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