 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2021/paper-1/15/b/solution

cosh x = 1 + \frac{x ^{2}}{2} + \frac{x ^{4}}{24} + O (x^{6}), cos x = 1 - \frac{x ^{2}}{2} + \frac{x ^{4}}{24} + O (x^{6}) .

Both are even, so write their quotient as

1 + a x^{2} + b x^{4} + O (x^{6})

a - 1/2

and

b - a /2 + 1/24

. Matching the numerator yields

a = 1

and

b = 1/2

. Thus

\frac{cosh x}{cos x} = 1 + x^{2} + \frac{x ^{4}}{2} + O (x^{6}) .

The denominator is nonzero near the expansion point, so division of the power series is valid there.

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