What are the most general and powerful math tricks?

Question

I quote from the first chapter of Michael Spivak's Differential Geometry Volume I:

The precise definition of $\mathbb{P}^2$ uses the same trick that mathematicians always use when they want two things that are not equal to be equal. The points of $\mathbb{P}^2$ are defined to be the sets $\{p,-p\}$ for $p \in S^2$...

That passage burned itself into my brain when I first read it decades ago. My immediate reaction was, "I want the list! Where is the list of tricks like that?!"

In my studies I have kept an eye out for tricks that generalize, that carry over into a different area. I also watch out for very broad categories of trick that have different specific versions in different topics. I tutor math and physics and have compiled lists of "debugging" steps for when a student is stuck on a problem. But I have long suspected that there are far more powerful tricks out there that are common knowledge to mathematicians but not to mathematics students.

In an effort to clarify what I am asking, I will write my own best answer, and I hope others can improve, elaborate, or best of all add completely new items to the list.

Answer 1

The most general math tricks I know include

1. Add zero. An example of this is completing the square. Faced with $x^2 + 6x + 1$, we can add $0 = 8-8$ to produce $x^2 + 6x + 9 - 8 = (x+3)^2 -8$.
2. Multiply by one. The simplest example of this is addition of fractions. The most common example is probably unit conversions. Other versions include the "conjugate trick" (see below).
3. Factoring an unknown. The solution to the cubic equation involves replacing the single variable $x$ with $u-v$, where you now have two variables, and at a later point can use the extra flexibility to declare some messy quantity to equal zero. A similar technique is variation of parameters in differential equations, where an unknown function is declared to be a different unknown function multiplied by a known one chosen for convenience.
4. Separation of variables. Independence in probability is one case, and another is in partial differential equations, writing the solution as a product of solutions depending on variables separately. Arguably that is a special case of #3.
5. The "Spivak trick" quoted in the question. I have rarely seen this used but it seems powerful and important.
6. The conjugate trick. A special case of #2, this is used extremely often, for rationalizing a denominator in fractions, complex arithmetic, L'Hopital's Rule, and more.
7. Symmetry. I don't even know how to characterize this. While I make significant use of it, there is often yet more symmetry that can be exploited to cut a problem in half, declare an answer zero, and so forth. It is clearly extremely powerful. But, for example, I once read a 12-page proof that a quintic equation does not have a general solution formula, which seemed to be 11.5 pages of group theory that seemed to have absolutely nothing to do with the question, followed by some handwaving and declaring "QED".
8. Solving a more general problem. This seems to require the most creativity, and so is hard to really classify as a tool more than a hope.

This is the broadest classification I know. I have constructed this mental model of techniques slowly over years, and I am curious what other powerful yet comprehensible techniques exist.