The invention of complex numbers allowed to express trigonometric functions through hyperbolic ones in closed form.

Is there possible an extension of real/complex numbers in which logarithms and inverse trigonometric functions can be expressed in terms of exponents/trigonometric functions and vise versa in closed form?

P.S. I have asked here but is seems people there just do not understand the question.

What I am talking about is something like this: $$\frac1\pi\ln \left(\frac{w-\frac{z}{\pi }}{w-1+\frac{z}{\pi }}\right)=\frac1z\cos (2wz)$$

or this:

$$\ln(\sin(w + z)) = 1/z \cos(2 w z)$$

Where $w$ is some element of the extended field, not a complex number. Is this possible?