Can someone prove cosh(z)=cos(iz)? thanks?

Answer 1

i presume you mean to say that z is a complex-valued variable here.

cosh(z) = [e^z + e^(-z)]/2

but e^(iz) = cos z + i sin z by the well-known Euler's Formula
therefore: cos z = [e^(iz) + e^(-iz)]/2 since the cosine function is an even function and the sine function is an odd function.

consequently: cos(iz) = [e^(-z) + e^z] / 2, since i^2 = -1 by definition of the imaginary unit i. your desired result follows from commutativity.

Answer 2

Write down the infinite series for cos x, then substitute iz for x. After you've simplified it (i^2=-1 etc), you should end up with the series for (1/2)(exp iz + exp -iz), which is cosh iz.