Proving Identities RigorouslyDate: 01/23/2002 at 18:35:19 From: Robert Chen Subject: Proving Identities This would probably apply to many of the proofs about identities, but since I was looking at identities, that'll be the topic. One submission in the Dr. Math archives was asking about (1-tan A)/sec A + (sec A/tanA) = (1+tan A)/(sec A tan A). One of the Doctor Maths answered it by saying to multiply both sides by sec A tan A. I do these two-sided operations frequently in math, and after I did VERY badly on my identities test, my teacher wrote "You can only work with one side at a time" beside my work. I was wondering why you work with both sides. I've been taught that the "proper" way is to work with one side only Date: 01/25/2002 at 00:02:39 From: Doctor Pete Subject: Re: Proving Identities Hi Robert, You bring up a very good point, and you are correct: the proper way to prove such identities is to begin on one side and algebraically transform it into the form shown on the other side. Working on both sides is technically incorrect because in doing so we are assuming that the equality to be proven is already true. However, in the case of many identities, it is allowable to work on both sides, in the sense that it is not mathematically incorrect - if the identity is true, then the result of working on both sides will eventually result in an equality that is "obviously" or more easily seen to be true. That is, such a method won't lead to a wrong decision on the truth or falseness of the identity in question. But in so far as mathematical rigor, it is insufficient because of the reason mentioned at the end of my first paragraph. Often, however, working on both sides can help us form a rigorous proof of an identity. I will illustrate this with the example identity you provided. Identity to be proven: (1-Tan[a])/Sec[a] + Sec[a]/Tan[a] = (1+Tan[a])/(Sec[a]Tan[a]) If we work on both sides, the first step is to multiply both sides by Sec[a]Tan[a]: (1-Tan[a])Tan[a] + Sec[a]Sec[a] = 1+Tan[a]. Multiplying through, we find Tan[a] - Tan[a]^2 + Sec[a]^2 = 1+Tan[a], or -Tan[a]^2 + Sec[a]^2 = 1, which is equivalent to the identity 1+Tan[a]^2 = Sec[a]^2, so we are done. This is not a rigorous proof, but it leads to one, in which we see the proper method is as follows: (1-Tan[a])/Sec[a] + Sec[a]/Tan[a] = ((1-Tan[a])Tan[a])/(Sec[a]Tan[a]) + Sec[a]^2/(Sec[a]Tan[a]) = (Tan[a] - Tan[a]^2 + Sec[a]^2)/(Sec[a]Tan[a]) = (Tan[a] - Tan[a]^2 + 1 + Tan[a]^2)/(Sec[a]Tan[a]) = (Tan[a] + 1)/(Sec[a]Tan[a]). Therefore we have taken the left-hand side and transformed it into the right-hand side using algebraic manipulation and other known trigonometric identities. But it should not be too hard to see that the basic "flow" of the proof is essentially the same procedure as the "improper" method, just rearranged a bit. Another, more sophisticated example of this principle is the proof of the following fact, known as the Arithmetic-Geometric Mean Inequality, for two numbers a, b >= 0: (a+b)/2 >= Sqrt[ab]. A common mistake is to use the two-sided method, like this: a + b >= 2 Sqrt[ab] (multiply both sides by 2) (a+b)^2 >= 4ab (square both sides) a^2 + 2ab + b^2 >= 4ab (expand the right side) a^2 - 2ab + b^2 >= 0 (subtract 4ab from both sides) (a - b)^2 >= 0 (factor the right side) and the last inequality is obviously true, since the square of any real number is never negative. But this is bad form, especially because we are dealing with an inequality. Therefore, the correct way is to begin with 0 <= (a - b)^2 = a^2 - 2ab + b^2 = a^2 + 2ab + b^2 - 4ab = (a + b)^2 - 4ab. Hence ((a+b)^2)/4 - ab >= 0, since if a number is greater than or equal to 0, that number divided by 4 is also greater than or equal to 0. Adding ab to both sides (we are allowed to do this because we already know that the inequality we began with is true), ((a+b)^2)/4 >= ab, and since a, b > 0, we may take the square root of both sides without affecting the inequality: thus (a+b)/2 >= Sqrt[ab], as was to be shown. In short, to be completely rigorous in our proof of trigonometric identities, we mustn't assume the identity, or in general, whatever needs to be proven, is true in the first place: it is much better to work on one side, trying to obtain the other, in a single chain of reasoning, and in other cases, working from already known identities (in which we may work on both sides) to obtain the desired result. - Doctor Pete, The Math Forum http://mathforum.org/dr.math/ |
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