I'm having trouble seeing how this piece of manipulation follows:
I can see that a factor of a 1/2 has been taken out but otherwise I'm unsure how to deal with the sinh(2arcsinh(wr))
Any help would be appreciated!
It's because $\displaystyle \displaystyle \sinh{2x} = 2\sinh{x}\cosh{x}$. In this case, $\displaystyle \displaystyle x = \arcsin{(\omega \tau)}$.
BTW, are you sure that it's $\displaystyle \displaystyle \sinh{[2\arcsin{(\omega \tau)}]}$ and not $\displaystyle \displaystyle \sinh{[2\,\textrm{arcsinh}\,{(\omega \tau)}]}$?