2sin(x+y)cos(x−y)

Question

Use the product-to-sum identities to rewrite the following expression as a sum or difference.

2sin(x+y)cos(x−y)= ??

Answer 1

There is a trig identity that states sin(a)cos(b) = .5[sin(a+b) + sin(a-b)], which we can use here.

Let a = x+y and b = x-y. Then we have,

sin(x+y)cos(x-y) = .5[sin((x+y) + (x-y)) + sin((x+y) - (x-y))].

Simplifying the right hand side, we get:

sin(x+y)cos(x-y) = .5[sin(2x) + sin(2y)]

Going back to the original question, we see that:

2sin(x+y)cos(x-y) = 2[sin(x+y)cos(x-y)] = 2[ .5[sin(2x) + sin(2y)]] = sin(2x) + sin(2y), which is the final answer.

Consider checking out this page from SOS math for more examples:

http://www.sosmath.com/trig/prodform/prodform.html

Answer 2

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

sin [(x + y) + (x - y)] + sin [(x + y) - (x - y)]

sin 2x + sin 2y

QED