 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/iii/solution

Write

cos x = 1 + u

with

u = - x^{2} /2 + x^{4} /24 - x^{6} /720 + O (x^{8})

. Substitution into

ln (1 + u) = u - u^{2} /2 + u^{3} /3 + O (u^{4})

gives

ln cos x = - \frac{x ^{2}}{2} + (\frac{1}{24} - \frac{1}{8}) x^{4} + (- \frac{1}{720} + \frac{1}{48} - \frac{1}{24}) x^{6} + O (x^{8}) .

Thus the first three nonzero terms of this even Taylor series are

ln cos x = - \frac{x ^{2}}{2} - \frac{x ^{4}}{12} - \frac{x ^{6}}{45} + O (x^{8}) .

The real logarithm is well defined near zero because

cos x > 0

there.

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