The preferred way to get aligned equations with numbering is amsmath package's align environment. See its documentation for help with that. It's quite simple, something like:
\begin{align}
f(x) & = \cos^2 x \\
g(x) & = \sin^2 x
\end{align}
There are a lot of variations trying to cover most conceivable equation alignment needs (again, check out the documentation).
As for your two-column proof format, I'm not as sure about the best way. A quick and dirty way would be to add it as a second column within the environment, something like:
\begin{align}
f(x) & = \cos^2 x & \text{this is the first function} \\
g(x) & = \sin^2 x & \text{this is the second function}
\end{align}
but this is no good for multi-line explanation, and puts the numbering to the right of the text. I'll try and think of a way (one that doesn't involve a lot of custom-defined environments, since surely someone's done this before).
Edit: As a starting point, this [sort of] works:
You can't do any alignment within the align environment (the & confuses things), and there are some vertical alignment issues - the align environment pads itself above and below, and the text in the right-hand cell. Maybe it's heading in a good direction, though!
\begin{tabular}{p{3 in}|l}
\begin{align} f(x) = \sin^2 x \end{align} &
this is the first equation \\
\begin{align} g(x) = \cos^2 x \end{align} &
this is the second equation
\end{tabular}