= Solution
The <hyperbolic functions> satisfy $\cosh^2\theta-\sinh^2\theta=1$, so a useful family is
$$
\boxed{a=\cosh\theta,\qquad b=i\sinh\theta,\qquad
A=\begin{pmatrix}\cosh\theta&i\sinh\theta\\-\varepsilon i\sinh\theta&\varepsilon\cosh\theta\end{pmatrix}.}
$$
For real $\theta$ this gives a one-real-parameter family of <complex orthogonal matrices>, which is enough to exhibit the requested hyperbolic parametrization. It does \b[not] parametrize every matrix in part (ii). For example $a=b=1/\sqrt2$ obeys the constraint but cannot have $a=\cosh\theta$ with real $\theta$, since then $a\ge1$.
Allowing complex $\theta$ gives the full parametrization: $(a-ib)(a+ib)=1$, so choose a <complex logarithm> $\theta$ with $e^\theta=a-ib$. Then $a=\cosh\theta$ and $b=i\sinh\theta$. A general complex pair therefore needs a complex parameter; the real-parameter restriction in the question should not be mistaken for a completeness claim.
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