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A section in my precalculus book is devoted to establishing (=proving) trigonometric identities, and a typical problem in the book presents a purported identity and asks students to establish it. The book recommends this method for doing so: Consider the more complicated-looking side of the purported identity. Use rules of algebra and known trigonometric identities to manipulate that side until it matches the other side. (Sometimes you'll need to manipulate both sides until they match one another.)

The book then has this warning:

Be careful not to handle identities to be established as if they were equations. You cannot establish an identity by such methods as adding the same expression to each side and obtaining a true statement. This practice is not allowed, because the original statement is precisely the one that you are trying to establish. You do not know until it has been established that it is, in fact, true.

Huh? I mean, I understand that you need to be careful. I understand that you can't manipulate your purported identity thus:$$\{\textrm{purported identity}\}\Rightarrow\{\textrm{something else}\}$$I understand that every implication must be instead like this:$$\{\textrm{purported identity}\}\Leftarrow\{\textrm{something else}\}$$And therefore, for example, one cannot raise both sides of the purported identity to an even power, or multiply both sides by $0$. Fine. But what's wrong with "adding the same expression to each side and obtaining a true statement"??

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marked as duplicate by msh210, kingW3, Ahaan S. Rungta, Lord_Farin, Matt Samuel Feb 18 '15 at 21:01

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  • $\begingroup$ When dealing with real numbers and the like, nothing is wrong with that since $a+b=a+c$ if and only if $b=c$ (just add $-a$ to both sides). So I disagree with your book in this context. Good of you to notice when you can't do the same thing to both sides though! $\endgroup$ – Hayden Feb 18 '15 at 13:55
  • $\begingroup$ There's nothing wrong with it, except that for clarity one should write it in the logical order : "A+x=B+x, therefore, A=B" (rather than "A=B because A+x=B+x"). It's a bit picky but helps keep things clear to the reader. $\endgroup$ – Sary Feb 18 '15 at 13:58
  • $\begingroup$ I suspect that the author isn't condemning stuff like $\text{Identity}\iff \text{True statement}$, therefore $\text{Identity}$ holds. But rather some other kind of mistake that students might regularly make. $\endgroup$ – Git Gud Feb 18 '15 at 14:12
  • $\begingroup$ It seems to me that the instructor is alluding to something like this : "When you are working with an identity, if you work on both sides and work down to where the sides are equal, you will only have shown that, if the starting equation is true, then you can arrive at another true equation. But you won't have proved, logically, that the original equation was actually true." $\endgroup$ – Mauro ALLEGRANZA Feb 18 '15 at 14:31
  • $\begingroup$ see here: math.stackexchange.com/questions/101053/… $\endgroup$ – hunter Feb 18 '15 at 15:01
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The author is trying to avoid the following error, which is typical of students beginning with proofs:

Proof that $-2 = 2$:

We have

$(-2)^2 = 2^2$

$4 = 4 \boxtimes$

However, as you have observed, sometimes we can reason from both sides of a purported equality, so long as we carefully argue that each of our steps is "reversible."

e.g.

Prove that $$ \cos^2(2x) + \sin^2(2x) + \tan(3x) = \tan(3x) + 1. $$ We observe that the equation holds if and only if the analogous equation holds with $\tan(3x)$ subtracted from both sides, but then this is obvious from the standard trig identities. That's a perfectly valid, complete proof.

So the book's insistence that you never do this is too strict, and will often serve to hamper creativity. I think it is a case where instructors of beginning proof-writing can harm good mathematical practice in order to prevent one error students make. I think it's better to play whack-a-mole with poor reasoning when it comes up than to broadly insist that one write a certain way. But other teachers disagree!

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When you try to prove something, you have to start with what you have and get to what you want.

Just as in any other aspect of life.

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  • $\begingroup$ This is wrong. Say you want to prove $a(x)=b(x)$. Then $a(x)=b(x)$ iff $a(x)+t = b(x) + t$ for all $x$. It doesn't matter with which equation you start since it is "$\Longleftrightarrow$". There are complications when squaring, but never adding or multiplying by a non-zero constant. $\endgroup$ – MathematicsStudent1122 Aug 8 '16 at 6:44
  • $\begingroup$ There are many complications —squaring is only one of them. Nothing cries «newbie» as much as a long column of equalities when trying to prove something. The advice given by the author of the book is correct, and what I wrote is a rephrasing of that advice. $\endgroup$ – Mariano Suárez-Álvarez Aug 8 '16 at 8:18

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