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The hyperbolic functions satisfy ${\cosh }^2\theta -{\sinh }^2\theta =1$, so a useful family isFor real $\theta$ this gives a one-real-parameter family of complex orthogonal matrices, which is enough to exhibit the requested hyperbolic parametrization. It does not parametrize every matrix in part (ii). For example $a=b=1/\sqrt{2}$ obeys the constraint but cannot have $a=\cosh \theta$ with real $\theta$, since then $a\ge 1$.

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.

Every two-by-two complex orthogonal matrix has form $A=\left(\begin{array}{cc}a& b\\ -\epsilon b& \epsilon a\end{array}\right)$, where $\epsilon =\det A\in \{+1,-1\}$ and $a^2+b^2=1$. This follows by equating $A^{-1}$ to $A^T$ in the two-by-two inverse formula, and direct multiplication proves the converse. A complete parametrization is $a=\cosh t$, $b=i\sinh t$ with complex $t$; restricting $t$ to real values selects only a subfamily.