When solving a trigonometric identity, is there any rule that says we can\'t man

Question

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?

Explanation / Answer

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.

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