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

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.