= Solution
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:
$$
\cos(a+x)=\cos a\left(1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots\right)
-\sin a\left(x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots\right).
$$
The two brackets are the <Maclaurin series> of cosine and sine, giving
$$
\boxed{\cos(a+x)=\cos a\cos x-\sin a\sin x.}
$$
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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