Simplifying Equivalent Functions Given two functions in closed form such that f(x) is the same for all x for both functions, is there always a way to manipulate either function to make it so they are written exactly the same or can you have two functions that can be proven equivalent yet neither can be simplified to look the same as the other?

For an explicit description of a situation where the identity problem is recursively unsolvable, see Richardson's Theorem.
Here the allowed operations are the constant function $\log 2$, the ordinary operations of arithmetic, $\exp$, $\sin$, absolute value, and composition of functions. There is no algorithm to determine whether an expression $f(x)$ built up using the single variable $x$ and the allowed operations is identically equal to $0$ on the reals.
