# 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?