I dont have a clue how people come up to the second part of the equation after the equal sign... please help me to understand!
I'm not exactly sure myself. If I wanted to prove the identity, I would do this:
$\displaystyle \displaystyle{\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}=\frac{e^{x}-e^{-x}}{2}\cdot\frac{2}{e^{x}+e^{-x}}=\frac{\sinh(x)}{\cosh(x)}=\tanh(x).}$
It depends on how you define the hyperbolic tangent function, I guess. Make sense?
$\displaystyle \displaystyle\frac{e^x}{e^x}=1$,
one of the countless ways to express the value 1.
Therefore...
$\displaystyle \displaystyle\frac{e^x-e^{-x}}{e^x+e^{-x}}=\frac{e^x-e^{-x}}{e^x+e^{-x}}\, . \frac{e^x}{e^x}=\frac{e^xe^x-e^xe^{-x}}{e^xe^x+e^xe^{-x}}=\frac{e^{x+x}-e^{x-x}}{e^{x+x}+e^{x-x}}$
giving a reduced form of tanh(x)
Hyperbolic function - Wikipedia, the free encyclopedia