I'd really appreciate someone telling me the right answer. My brain is way to tired to figure it out (or understand an explanation).
Thanks
#1 and #3 are proven correctly.
To see why #2, and #4 are wrong:
For #2: $\displaystyle \frac{1}{2}[e^{-x} + e^{-(-x)}] \neq \frac{1}{2}(e^{-x} - e^x)$
For #4: $\displaystyle \frac{1 + \tanh{x}}{1 - \tanh{x}} = \frac{\cosh{x} + \sinh{x}}{\cosh{x} - \sinh{x}} \neq \frac{\cosh{x} - \sinh{x}}{\cosh{x} + \sinh{x}}$
You may want to refer to: Hyperbolic function as these proofs are very easy to verify.