= Solution
The <Maclaurin series> of the <hyperbolic cosine> and <cosine> are
$$
\cosh x=1+\frac{x^2}{2}+\frac{x^4}{24}+O(x^6),\qquad\cos x=1-\frac{x^2}{2}+\frac{x^4}{24}+O(x^6).
$$
Both are even, so write their quotient as $1+ax^2+bx^4+O(x^6)$. Multiplication by the denominator gives coefficients $a-1/2$ and $b-a/2+1/24$. Matching the numerator yields $a=1$ and $b=1/2$. Thus
$$
\boxed{\frac{\cosh x}{\cos x}=1+x^2+\frac{x^4}{2}+O(x^6).}
$$
The denominator is nonzero near the expansion point, so division of the <power series> is valid there.
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