Question

Combine the sum and difference identities of sine to show how 1/2[sin(x+y)+sin(x-y)] equals sin(x)cos(y).

Answer 1

To combine the sum and difference identities of sine, we can use the trigonometric identity: sin(u) + sin(v) = 2sin((u+v)/2)cos((u-v)/2). Now let's substitute u = x+y and v = x-y into the identity. Therefore, 1/2[sin(x+y)+sin(x-y)] equals sin(x)cos(y).

Step-by-step explanation:

To combine the sum and difference identities of sine, we can use the trigonometric identity: sin(u) + sin(v) = 2sin((u+v)/2)cos((u-v)/2).

Now let's substitute u = x+y and v = x-y into the identity.

1/2[sin(x+y)+sin(x-y)] = 2sin((x+y+x-y)/2)cos((x+y-x+y)/2) = sin(x)cos(y). Therefore, 1/2[sin(x+y)+sin(x-y)] equals sin(x)cos(y).