Proving the Complex Hyperbolic Property Using Trigonometric Construction

[nichkis]

Homework Statement

Show: sinh(z + i2(pi)) = sinh(z) using sinh(z) = (e^z - e^-z)/2

Homework Equations

The Attempt at a Solution

So far I have (e^{x + i(2∏+y)} - e^{-(x+i(2∏+y)})/2.

Need help proceeding from here. My thoughts were to define a z' = x + i(2∏+y) but I don't think that I can then say that it is equal to sinh(z) but rather sinh(z').

 

[Dick]

What is e^{2 \pi i}? Then use the laws of exponents.

 

[nichkis]

Using the trigonometric construction I was able to get to the answer. Thanks a bunch.