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### Proving Identities Rigorously

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Date: 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.

(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
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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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Associated Topics:
College Trigonometry
High School Trigonometry

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