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{x\cos 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?