I am reading a book where i have come across an expression:
$$ \frac{e^{x}\left(-a^2+b^2+2iab\right)- e^{-x}\left(-a^2 +b^2-2iab\right)}{2} $$
In the book they state that we can express this equation in terms of hyperbolic sin like this:
$$ \sinh x\cdot \left(-a^2+b^2\right) $$
It is not so obvious to me how they do this although i know the definition of hyperbolic sine...
Then they just rewrite first equation like this:
$$ \frac{e^{-x}\left(a^2 -b^2+2iab\right) +e^{x}\left(-a^2+b^2+2iab\right)}{2} $$
I understand this, but then again they express the above in terms ofhyperbolic cosine like this:
$$ \cosh x\cdot \left(2iab\right) $$
Again, it is not obvious to me how they do this... Could someone please point that out?
As $$\sinh(x)=\frac{e^x-e^{-x}}2,\cosh(x)=\frac{e^x+e^{-x}}2,$$
$$ \frac{e^{-x}\left(a^2 -b^2+2iab\right) +e^{x}\left(-a^2+b^2+2iab\right)}{2} $$
$$=-(a^2-b^2)\left(\frac{e^x-e^{-x}}2\right)+2abi\left(\frac{e^x+e^{-x}}2\right)$$
$$=-(a^2-b^2)\sinh(x)+2abi \cosh(x)$$