can anyone give the answer?...this question has a paradox

Question

Find $\cos(z)$, given $$z=i\ln(2-\sqrt3))$$

Answer 1

Note,

$$\cos z = \cos( i\ln(2-\sqrt3))=\cosh(\ln(2-\sqrt3))=\frac12(2-\sqrt3 + \frac1{ 2-\sqrt3 })=2$$

So, it is actually the range of $\cosh(z)$ after converting to real space.

Answer 2

The range of $cos(x)$ is in [-1 .. 1] on the real line only. When extended to the complex plane, $cos(z)$ has a complex range , which goes outside that interval for non-real inputs.