Confusion with product-to-sum trig identities

[Jyan]

I'm having some confusion with a couple trig identities. On wikipedia (http://en.wikipedia.org/wiki/List_of_trigonometric_identities#Product-to-sum_and_sum-to-product_identities), the following two identities are listed:

sinθcosβ = (1/2)[sin(θ+β) + sin(θ-β)]

and

sinβcosθ = (1/2)[sin(θ+β) - sin(θ-β)]

I can see the difference between them if they are used with the same variables θ and β. But, how do you know which one is valid in any given situation? I find this a hard question to phrase, but I hope you can see my confusion. If you have sin x cos y, which identity can you apply? Does it matter? So long as you apply the other one to sin y cos x?

 

[Ackbach]

They're really the same identity - one of them being superfluous. If you swap the roles of the arguments, and use the fact that $\sin(-x)=-\sin(x)$, then you will see that they are the same.

 

[Jyan]

I see, thank you.

 

[Ackbach]

You're welcome!