I’ve ranted in the past about the fallacy of trying to prove an identity by starting with the equation itself, then manipulating both sides of the equation until you arrive at a valid identity.
While grading some homework this term (involving proofs of trig identities), I found the need to raise the subject again in class. But my stock example, proving that -3 = 3 by squaring both sides, seemed too transparent. I wanted something where the fallacy was solely due to proving that False implies True.
I ended up with the following example, which I like a lot, but which I’m certain has been rediscovered by others over the ages. Still, it’s a good illustration of why we can’t prove identities in this way.
Claim: , for all .
Proof: Assume that . If we square both sides, this implies that . Furthermore, since equality is reflexive symmetric, it follows that .
Finally, adding these two equations gives us , which reduces to the equation 1=1. QED.
The careful reader will note that squaring both sides is irrelevant, as is the Pythagorean identity for sine and cosine. In essence we have a general proof that for any expressions and : If , then by reflexivity symmetry we know that , and thus . But since addition is commutative, this reduces to the identity .
I prefer the slight obfuscation of the proof over the distilled simplicity of “reflexive symmetric plus commutative”.
Your mileage may vary.