Tips and tricks for proofs involving algebraic / equation manipulation I recently completed a solid chunk of my first truly formal math book on Set Theory and decided to move on to Michael Spivak's Calculus (as I heard it was a good place to begin one's journey into Analysis). For the purpose of self-studying, I purchased the solution manual, as well.
After completing the first two chapters and going through all practice problems, I am now painfully aware that I am not particularly great at proofs involving clever equation manipulations. As I progressed through my aforementioned set theory book, I felt like the number of "proof flavors" (different strategies one could invoke to tackle the claim) were quite manageable. Figuring out which path to pursue  seemed to be categorically recognizable.
However, when it comes to proofs involving equation manipulation, I literally feel like I am staring into a sea of hundreds of possible permutations (in the English sense) that one might use to arrive at the desired conclusion. From what I have seen so far, the strategies largely seem to involve the invocation of Field Axioms (e.g. creative usages of $0$ and $1$ through the existence of additive inverses and  multiplicative inverses, respectively) and factoring (that often times feels quite contrived).
I was wondering if there were any general tips and tricks as to how one might improve his/her ability in recognizing the type of equation manipulation that a given proof is suggesting (yelling) to carry out. I am assuming the seasoned mathsters have some sort of pattern recognition that goes off in their heads that allows for efficient determination of starting points.
I apologize if this question is a little too broad. I recognize that every proof may necessarily be "special", so in the absence of a specific example, perhaps this question is unanswerable. But here's hoping. Book recommendations are also greatly appreciated! Cheers~
 A: I'm about halfway through Spivak's Calculus myself.
The specific problem you mention (Prove $\sqrt 6-\sqrt 2 - \sqrt3$ is irrational) is a tricky one. I certainly wasn't able to do it myself. I ended up having to look at the published solution. Coming back to it today, several months later though, I managed it.
(Note: the next part of that problem ("prove that $\sqrt2$ + $\sqrt[3]{2}$ is irrational") is marred by a hint ("start by working out the first 6 powers of this number") that's meant to push you towards the published solution, which relies on linear algebra and is certainly outside the scope the book. Usually Spivak is loath to rely on things without justification. This is an unusual lapse, made more baffling by the existence of a much more appropriate solution.  Compounding things, I think there's also a typo in the table in his solution. There are mistakes in the text, ranging from typos to more serious problems. When you think there might be a mistake, MSE can be very helpful.)
More generally, many of the problems in the book are challenging. Often, I have to look up the solution, at least for a hint.
The difficulty is also somewhat front-loaded. The first 4 Chapters which in 3rd edition are "Basic Properties of Numbers", "Numbers of Various Sorts", "Functions", and "Graphs" (with 3 appendices), were in some ways more challenging for me than anything that's come since.
Once you hit the first chapter on "Limits", things settle into a groove.
Not that it's "easy" from that point on; the problems remain challenging throughout. Just that, I think you'll find from limits on, it gets really fun.
Don't be discouraged if you aren't able to "ace" the problems. Many of the early problems are introducing you to concepts and techniques you will use later on. The full import of some things doesn't become clear until many chapters later.
You can learn most of Spivak's clever tricks in Spivak, as opposed to trying to learn them elsewhere ahead of time just in case they are in Spivak.
Certainly look outside of Spivak if you need to brush up on something specific, but I don't know if it's worth it trying to just generally get better at clever arithematic tricks before moving on with the text.
Remain diligent about doing the problems. If you make sure you understand them, even the ones you need help with, you will get a lot out of it.
Go back periodically and try re-working problems that initially gave you a hard time.
The book is a tough, glorious, frustrating, edifying beast. Stick with it at least until the end of the "Limits" chapter. I think at that point you'll be happy you did.
