When solving a trigonometric identity, is there any rule that says we can't manipulate the left side/right side - bring terms across the equal sign - before we start - as long as at the end of it...

When solving a trigonometric identity, is there any rule that says we can't manipulate the left side/right side - bring terms across the equal sign - before we start - as long as at the end of it all, LS = RS?

For instance, If we had

2cos(x) + 1 = sin(x)

Could we bring the 1 across the equals sign to make it:

2cos(x) = sin(x) - 1

Before we actually start using our identities?

Note, the above [2cos(x) + 1 = sin(x)] is simply just an example intended to better illustrate the question - I'm not sure if it even an identity.

Also, is there more then one correct way to solve a trigonometric identity? There must be right?

Asked on by lroster

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embizze's profile pic

embizze | High School Teacher | (Level 1) Educator Emeritus

Posted on

First, I agree with degeneratecircle on the correct use of terminology. Solving an equation and proving an identity are different.

Next, if solving an equation it is permissable to make valid algebraic "moves", e.g. add the same thing to both sides, etc...

However, when proving an identity this is not correct. If you want to prove `cos(2x)=cos^2x-sin^2x` this is not the same as proving `cos(2x)+sin^2x=cos^2x` . If the first equation is an identity, then the second equation will be valid. However, if the first equation is not an identity, then whatever you might prove is invalid.

By adding `sin^2x` to both sides, you assumed that the sides of the equality were equal, but that is what you are trying to prove. This is a circular argument, and not a valid proof technique.

The only valid proof is to start with one side, and with valid manipulations get to the other side. That is how the final product appears. But in practice, you often work on both sides until you can show the sides are equal.

`cos(2x)=cos(x+x)`

`=cosxcosx-sinxsinx`

`=cos^2x-sin^2x`

would be a correct proof.

Note that there may be many different ways to prove the identity. There are books written with nothing but proofs of the Pythagorean theorem. Identity proofs are just like geometric proofs in that regard.

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degeneratecircle's profile pic

degeneratecircle | High School Teacher | (Level 2) Associate Educator

Posted on

I think maybe I wasn't clear in my first post. I wasn't suggesting that we use `cos(2x)+sin^2x=cos^2x` to prove that `cos(2x)=cos^2x-sin^2x,` since they're the same statement just with one term brought to the other side.

I was saying that if you can prove that `cos(2x)+sin^2x=cos^2x` is an identity (by starting with one side and ending on the other), then it follows that `cos(2x)=cos^2x-sin^2x.`

degeneratecircle's profile pic

degeneratecircle | High School Teacher | (Level 2) Associate Educator

Posted on

Before I answer, just a minor vocabulary correction: usually we either "solve an equation" or "prove an identity", but not "solve an identity". Since identities are true for all values of `x` where both sides are defined, the solution to them is simply all values of `x` where both sides are defined. However, in order to say that this is the solution, we first have to prove the identity.


Anyway, you're right that if you were to solve the equation `2cosx+1=sin x,` or prove the identity, say, `cos(2x)=cos^2x-sin^2x,` then you could instead solve `2cosx=sinx-1` or prove `cos(2x)+sin^2x=cos^2x,` respectively, if those are easier for you.

You're also correct that there is usually more than one way to solve an equation or prove an identity, and the more advanced it is, often times the more ways there are to approach it because you'll have proved many other identities beforehand. See the link for some proofs of ``the angle addition formulas in trig.

Sources:

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