Is there a function to convert a trigonmetric expression to an algebraic expression suitable for ComplexPlot?

According to H. A Priesley:

"both $cosθ$ and $sinθ$ can be written as simple algebraic functions of
$z$":

$cosθ =frac{1}{2} (z + frac{1}{z}) tag1$
$sinθ =frac{1}{2i} (z − frac{1}{z})tag2$

Which is very handy but I’m not sure what function will convert say $frac{sinh(ax)}{sinh(bx)},$ ot $ frac{xcos ax}{1+x^2}coth frac{pi x}{4}$ to similar algebraic functions. The only function I can find is TrigtoExp, but this does not include the needed substution $ z = exp iθ$ to work with function ComplexPlot.

Q1. Is there a function to convert a trigonmetric expression to an algebraic expression suitable for ComplexPlot?

Q2. If not what is the best way to get around this?