= Solution
The <Maclaurin series> of the <hyperbolic cosine> is $\cosh x=1+x^2/2+x^4/24+O(x^6)$. The <binomial series> with exponent $-1/2$ gives $(1+u)^{-1/2}=1-u/2+3u^2/8+O(u^3)$. Substituting $u=x^2/2+x^4/24+O(x^6)$ yields
$$
\begin{aligned}
(\cosh x)^{-1/2}&=1-\frac12\left(\frac{x^2}{2}+\frac{x^4}{24}\right)+\frac38\left(\frac{x^2}{2}\right)^2+O(x^6)\\
&=\boxed{1-\frac{x^2}{4}+\frac{7x^4}{96}+O(x^6).}
\end{aligned}
$$
These are the first three nonzero terms; the remainder is written in <Big O notation>.
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