Osborne's rule for hyperbolic functions?

Question

I am confused as to why you only change the sign for powers of sine that are 4n+2.

As I understand,

$sin(i\theta)=isinh(\theta)$

$sin^2(i\theta)=-sinh^2(\theta)$

$sin^3(i\theta)=-isinh^3(\theta)$

$sin^4(i\theta)=sinh^4(\theta)$

So I would think that the third power would also have the sign changed, although I am also slightly confused about what difference it makes whether there is an i present or not...

So I have 2 questions:

1. What is the significance of the i when you have odd powers of sinh? How does it affect the relationship between $sin(i\theta)$ and $sinh(\theta)$?

2. Why is it that the sign change only applies for powers of sine that are 4n+2?

I have read this post but it really has not clarified very much for me. Explanations that use fewer technical terms would be much appreciated!

Thank you :)

Answer 1

You have to also look at when 4n+1 powers of sinh occurs, for example if it only occurs when expressing sinh(a+b) then you have to divide by i once therefore leaving the identity unchanged. For 4n+2 if it only occurs with identities involving cosh(a+b) you don't divide because cos(ix) = cosh(x). for similar reasons with 4n+3 you divide by -i. This rule can be proven with individual cases but imo they require far too much unnecessary thinking to describe using some sort of intuition