I have been working on this one for a couple of hours and i just get stuck on every attempt i make.
I have to reduce the formula algebraically:
$\sinh(2 \cdot \sinh^{-1}(y))$
And I just can't seem to do it. I tried using the hyperbolic addition formulas to do something but I just ended up with an even more convoluted expression.
I tried using the addition formula with
$\sinh(x + x) = \cosh(x)\sinh(x) + \sinh(x)+\cosh(x)$
where $x$ is $\sinh^{-1}(y)$,
and then I replaced $\cosh(x)$ and $\sinh(x)$ with their definitions. It did not work.
Can anyone help me out here?
Your idea of using the identity would actually work.
$\sinh(2x) = 2 \sinh(x) \cosh(x)$ where $x = \sinh^{-1}(y)$
$= 2 y \cosh(x)$.
Now $\cosh(x) = \sqrt{1 + \sinh(x)^2}$.
Can you finish?