How is the following statement true?

Question

I am struggling to prove the statement:$$\sinh\left(\pi\left(\frac12+m\right)i\right)=(-1)^mi$$($m\in\mathbb Z$). I have tried using the exponential form of $\sinh$ and then changing back to $\cos$ and $\sin$ making use of the imaginary unit, but it doesn't seem to work since I get $ i\sin(\pi(1/2+m)) $ and since $(1/2+m)=2m+1=$odd for any integer $m$ then $\sin$ of this value is 0 rather than $(-1)^m$.

Answer 1

Please note that $i\sin\left(\pi\left(\frac12+m\right)\right)=i\sin\left(\pi\frac{2m+1}2\right)=i\sin\left(m\pi+\frac\pi2\right)=i(-1)^m$.