Why can't I cross multiply when proving trig. identities?

Question

I lost six marks on a test because of this. Is it because it negates the restrictions that the denominators carry? or is it some rule created by mathematicians/my teacher?

Answer 1

The basic rule is that you have to consider LHS and RHS of equation separately.
Have then to prove that LHS = RHS

Answer 2

Proving Trig Identities

Answer 3

You'll have to show an example of what you mean but if you mean cross multiplying in the sense that

a/b = c/d --> ad = bc

then you are able to do so. You just have to note the values where the denominators are 0 and exclude them from your domain.

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Those who are suggesting that cross multiplication or any other valid mathematical operation is not valid in the context of a proof are wrong.

The use of similar tools, such as proof by contradiction, has been part of mathematics for millenia.

In the case of proving an trigonometric expression, you are proving whether it is true, not making a statement. So of course you can use valid mathematical operations such as cross multiplication within the proof.

As long as the use of valid mathematical operations brings the original expression to a clear true (1=1) or false (1=2) result, then you have made a statement about the truth or falsity of the original expression which was the goal.

This is why I find statements that "well you don't know if they are equal or not so you can't assume they are" so absurd because that is the whole point of the exercise!

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Edit: When you "cross multiply" you are not actually mixing up the LHS and the RHS. You are just multiplying both sides by the same expression which just happens to be the product of the denominators.

a/b = c/d
a/b * bd = c/d * bd
ad = bc

There is no special exemption such that you can multiply each side by the same expression as long as its not the product of the denominators.

Answer 4

If two fractions are equal such as 2/3 = 4/6, the the cross products are equal: 2 x 6 = 3 x 4.

But when you are trying to prove an identity you do not know the two expressions are equal. In fact they may not be equal and you will end up proving there is no identity. If you cross multiply you are assuming the expressions are equal. If you are going to assum they are equal it is not necessary to prove they are equal. But assumpitons can be wrong.

This would be clearer if the instructions for proving identites were stated as follows: Some of the equations below are true and therefore identities, but some are false and not identities. Without assuming they are true or false, determine which are true and which are false.

Answer 5

Here is what I will paraphrase from a website:

The equal sign has NOT already been proven. YOU are proving that they are equal and can not, therefore, make the assumption that they are equal and that cross multiplying will work.You must work on each side individually first to get a common form. I'm sorry you lost points, but next time, you won't!

Answer 6

It has to do with the restrictions of the denominators. You can't cross-multiply because then, the answer wouldn't be accurate especially if the denominator is a function.