The approach you used is more common than you think (i.e. choosing a different way to prove something rather than the intended way).
People introduce all kinds of transformations and substitutions (like adding a - a terms in) so it may actually be the case that for a result, someone has added extra terms and then shown equivalence to an existing result (like the example you have shown).
Unless you specifically have to start out with the LHS, I wouldn't worry about it (and for the future, keep in mind of how introducing new variables and substitutions is useful for showing equivalence between multiple representations of the same thing).
Hi chiro, yup I try to take note of the different substitutions but this question was actually part of an overall problem where I had to go specifically form the LHS to the RHS. After I had a sneak peek of the solution (which is the RHS), I then backtracked to the LHS which convinced myself that the two expressions are indeed equal, however without sneak peeking at the final answer, there would have been no way for me to go from the LHS to the RHS, I'm wondering if there are any clear methods to manipulate the LHS to the RHS, strictly.