Observe the following facts; $\theta$ will be a real parameter throughout. From the definition $\cos z = \frac12 (e^{iz} + e^{-iz})$ and $\sin z = \frac1{2i} (e^{iz} - e^{-iz})$ for $z\in\mathbb{C}$. So
$$ \cos i\theta = \frac12 (e^{-\theta} + e^{\theta}) = \cosh \theta $$
and
$$ \sin i\theta = \frac1{2i} ( e^{-\theta} - e^{\theta}) = i\sinh\theta $$
Now, notice that $\cos$ and $\sin$ are holomorphic functions when looked at as functions on the complex plane. And a trigonometric identity can be expressed as
$$ R(\cos \theta,\sin\theta, \cos 2\theta, \sin 2\theta, \ldots, \cos k\theta, \sin k\theta) = Q(\cos\theta,\sin\theta) $$
where $R$ and $Q$ are rational functions (functions expressible as a polynomial divided by another polynomial) of suitable number of variables, this implies that a trigonometric identity is asserting that two meromorphic functions on the complex plane take the same values when restricted to the real axis. Now by a fundamental property of meromorphic and holomorphic functions, this implies that the two meromorphic functions are in fact everywhere equal: in particular they are equal on the imaginary axis. That is, we have
$$ R(\cos i\theta, \sin i\theta, \ldots) = Q(\cos i\theta,\sin i \theta) $$
which by the identities above can be written as
$$ R(\cosh \theta, i\sinh\theta, \ldots) = Q(\cosh \theta, i\sinh\theta) $$
thus showing Osborne's rule.
