Past exam of the mathematical methods II course of the University of Oxford / 2017 resit / 1

The successive derivatives of cosine cycle through $\cos a,-\sin a,-\cos a,\sin a$. Its Taylor series about $a$ therefore separates into even and odd powers:The two brackets are the Maclaurin series of cosine and sine, givingBoth series converge absolutely for every real $x$, so regrouping the even and odd terms is valid everywhere on the real axis.

The hyperbolic cotangent isExpand the numerator and denominator using the exponential Maclaurin series:This is a Laurent series, rather than an ordinary Maclaurin series: $\coth x$ has a simple pole at zero. The printed request should be understood as the expansion obtained from the exponential series, including this unavoidable $x^{-1}$ term.