Verify that each equation is an identity.
sin(x + y) + s...

Question

Verify that each equation is an identity.

sin(x + y) + sin(x - y) = 2 sin x cos y

Accepted Answer

Hello, everyone. We are asked if the given equation is an identity. We are given the sine of X minus Y minus the sine of X plus Y equals negative two sine, Y cosine X. Our answer choices are a yes, B no. So when we're asked if this is an identity, we basically want to know does the left hand side equal the right hand side. So I'm gonna start by simplifying the left hand side to see if I can get it to match the right hand side. So if we have the sine of X minus Y minus the sign of X plus Y, I'm gonna use a few identities. So I know that the sine of X minus Y equals the sine of X cosine of Y minus the cosine of X sine of Y from our different identity and from the sum of Engel's sign identity, I know that the sine of X plus Y will equal the sine of X cosine of Y plus the cosine of X sine of Y. So I'm going to plug this in to what we were given on the left hand side of our equation. So I will have the sine of X cosine of Y minus the cosine of X sine of Y minus. And I'm gonna make a square bracket. The sine of X cosine of Y plus the cosine of X sine of Y. I put it in a bracket. So that way I could remember to distribute my negative sign into it. And that's what I'm doing in my next step. So I'll have the sine of X co sine of Y minus the cosine of X sine of Y minus the sine of X cosine of Y minus the cosine of X sine of Y. So now let me look for like terms. I have the sine of X cosine of Y and I have a negative sine of X cosine of Y. Those are opposites. So they will cancel and I'm left with negative cosine of X sine of Y minus the cosine of X sine of Y. This could be rewritten as negative two cosine of X sine of Y. And let's see, does that match our right hand side? On the right hand side, we had negative two sine of Y cosine of X. Well, in multiplication order does not matter. So we could say that is the same as negative two cosine of X sine of Y. So yes, answer choice. A this is ID, an identity have a nice day.

Struggling with Trigonometry?

Verify that each equation is an identity.
sin(x + y) + sin(x - y) = 2 sin x cos y

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Sum and Difference Identities practice set

Struggling with Trigonometry?

Verify that each equation is an identity.sin(x + y) + sin(x - y) = 2 sin x cos y

Watch next

Sum and Difference Identities practice set

Verify that each equation is an identity.
sin(x + y) + sin(x - y) = 2 sin x cos y