Can you use both sides of an equation to prove equality?
For example:
$\color{red}{\text{Show that}}$$$\color{red}{\frac{4\cos(2x)}{1+\cos(2x)}=4-2\sec^2(x)}$$
In high school my maths teacher told me
To prove equality of an equation; you start on one side and manipulate it algebraically until it is equal to the other side.
So starting from the LHS: $$\frac{4\cos(2x)}{1+\cos(2x)}=\frac{4(2\cos^2(x)-1)}{2\cos^2(x)}=\frac{2(2\cos^2(x)-1)}{\cos^2(x)}=\frac{4\cos^2(x)-2}{\cos^2(x)}=4-2\sec^2(x)$$ $\large\fbox{}$
At University, my Maths Analysis teacher tells me
To prove a statement is true, you must not use what you are trying to prove.
So using the same example as before:
LHS = $$\frac{4\cos(2x)}{1+\cos(2x)}=\frac{4(2\cos^2(x)-1)}{2\cos^2(x)}=\frac{2(2\cos^2(x)-1)}{\cos^2(x)}=\frac{2\Big(2\cos^2(x)-\left[\sin^2(x)+\cos^2(x)\right]\Big)}{\cos^2(x)}=\frac{2(\cos^2(x)-\sin^2(x))}{\cos^2(x)}=\bbox[yellow]{2-2\tan^2(x)}$$
RHS =$$4-2\sec^2(x)=4-2(1+\tan^2(x))=\bbox[yellow]{2-2\tan^2(x)}$$
So I have shown that the two sides of the equality in $\color{red}{\rm{red}}$ are equal to the same highlighted expression. But is this a sufficient proof?
Since I used both sides of the equality (which is effectively; using what I was trying to prove) to show that $$\color{red}{\frac{4\cos(2x)}{1+\cos(2x)}=4-2\sec^2(x)}$$
One of the reasons why I am asking this question is because I have a bounty question which is suffering from the exact same issue that this post is about.
EDIT:
Comments and answers below seem to indicate that you can use both sides to prove equality. So does this mean that my high school maths teacher was wrong?
$$\bbox[#AFF]{\text{Suppose we have an identity instead of an equality:}}$$ $$\bbox[#AFF]{\text{Is it possible to manipulate both sides of an identity to prove that the identity holds?}}$$
Thank you.
Solution 1:
There's no conflict between your high school teacher's advice
To prove equality of an equation; you start on one side and manipulate it algebraically until it is equal to the other side.
and your professor's
To prove a statement is true, you must not use what you are trying to prove.
As in Siddarth Venu's answer, if you prove $a = c$ and $b = c$ ("working from both sides"), then $a = c = b$ by transitivity of equality. This conforms to both your teacher's and professor's advice.
Both your high school teacher and university professor are steering you away from "two-column proofs" of the type: \begin{align*} -1 &= 1 &&\text{To be shown;} \\ (-1)^{2} &= (1)^{2} && \text{Square both sides;} \\ 1 &= 1 && \text{True statement. Therefore $-1 = 1$.} \end{align*} Here, you assume what you want to prove, deduce a true statement, and assert that the original assumption was true. This is bad logic for at least two glaring reasons:
If you assume $-1 = 1$, there's no need to prove $-1 = 1$.
Logically, if $P$ denotes the statement "$-1 = 1$" and $Q$ denotes "$1 = 1$", the preceding argument shows "$P$ implies $Q$ and $Q$ is true", which does not eliminate the possibility "$P$ is false".
What you can do logically is start ("provisionally", on scratch paper) with the statement $P$ you're trying to prove and perform logically reversible operations on both sides until you reach a true statement $Q$. A proof can then be constructed by starting from $Q$ and working backward until you reach $P$. Often times, the backward argument can be formulated as a sequence of equalities, conforming to your teacher's advice. (Note that in the initial phase of seeking a proof, you aren't bound by anything: You can make inspired guesses, additional assumptions, and the like. Only when you write up a final proof must you be careful to assume no more than is given, and to make logically-valid deductions.)
Solution 2:
It is enough.. Consider this example:
To prove: $a=b$
Proof: $$a=c$$ $$b=c$$ Since $a$ and $b$ are equal to the same thing, $a=b$.
That is the exact technique you are using and it sure can be used.
Solution 3:
To prove a statement is true, you must not use what you are trying to prove.
The main problem is that you've misunderstood what the second teacher said (and possibly the meaning of the equals sign). What that second teacher is saying is do not take as prior facts what you're trying to prove.
As one textbook puts it (Setek and Gallo, Fundamentals of Mathematics, 10th Edition, Sec. 3.8), "An argument, or proof, consists basically of two parts: the given statements, which are called premises, and the conclusion". Wikipedia says this (sourced from Cupillari, Antonella. The Nuts and Bolts of Proofs. Academic Press, 2001. Page 3.): "In mathematics and logic, a direct proof is a way of showing the truth or falsehood of a given statement by a straightforward combination of established facts, usually axioms, existing lemmas and theorems, without making any further assumptions."
Now consider a simple equation/identity like $6 = 2 \times 3$. The separate sides are expressions; if you just said "6" in English that's a sentence fragment, not an assertion of any fact. It can't be evaluated as either "true" or "false", because it has no assertive content. It cannot be used as a premise because it's not a proposition.
What makes something a fully-formed statement in mathematical language is a relation, most commonly equals (but alternatively "is lesser than", "is greater than", etc., effectively the verbs of the language). Translating the equation $6 = 2 \times 3$ to English we get "6 is the same as 2 times 3", which is indeed a full sentence. This can be checked as being true or false; it makes an assertion. It can be used as a premise because it is a proposition of a particular fact.
In conclusion, both your teachers are correct, and both of your proofs are correct (although most of us would prefer the more concise one). When one says "don't use what you're trying to prove" they're not talking about the appearance of any particular expression in an algebraic transformation; expressions are neither premises nor things that can be proven; they are sentence fragments. They're talking about an assertion of fact, which in math has to be a statement including a relational symbol (most commonly an equation). The fact that you didn't start by assuming that equality means that in both cases you've complied with your second teacher's warning.