 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2017-resit/1/a/solution

The successive derivatives of cosine cycle through

cos a, - sin a, - cos a, sin a

a

therefore separates into even and odd powers:

cos (a + x) = cos a (1 - \frac{x ^{2}}{2 !} + \frac{x ^{4}}{4 !} - \dots) - sin a (x - \frac{x ^{3}}{3 !} + \frac{x ^{5}}{5 !} - \dots) .

cos (a + x) = cos a cos x - sin a sin x .

Both series converge absolutely for every real

x

, so regrouping the even and odd terms is valid everywhere on the real axis.

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