= Solution
Divide numerator and denominator by $x$ before taking their <Maclaurin series>:
$$
\frac{\sin x}{\sinh x}
=\frac{1-x^2/6+x^4/120+O(x^6)}{1+x^2/6+x^4/120+O(x^6)}.
$$
For the reciprocal denominator, the <geometric series> gives $1-x^2/6+7x^4/360+O(x^6)$. Multiplying gives the <Taylor expansion>
$$
\boxed{\frac{\sin x}{\sinh x}=1-\frac{x^2}{3}+\frac{x^4}{18}+O(x^6).}
$$
Strictly, the original quotient has a <removable singularity> at zero; this is the <Taylor series> of its continuous, analytic extension with value one there. The absence of odd powers also follows because the ratio of these two odd functions is even.
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