Question 1100909: sin(x)*cos(y)=1/2[sin(x+y) + sin(x-y)] (1)
cos(x)*sin(y)=1/2[sin(x+y) - sin(x-y)] (2)
It reads: verify the product-to-sum identities below using the sine sum and difference identities.
I have no clue where to begin. This is an online course and the instructor has disabled the "show me an example" option and the "help me solve" option. I want to learn this and if you guys can teach me It would be appreciated. Thanks in advance.
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
The sum and difference identities for sine are these equations below
sin(x+y) = sin(x)cos(y)+cos(x)sin(y)
sin(x-y) = sin(x)cos(y)-cos(x)sin(y)
These identities are often found in the appendix of your trig textbook, and also in the relevant chapters of the text. Depending on your teacher, you will either have a reference sheet, they will be given in the problem, or you have to memorize these formulas.
Add the equations shown above. We do so by adding the left sides separately and then the right sides separately. On the left side we will get sin(x+y)+sin(x-y). On the right side we will have
[ sin(x)cos(y)+cos(x)sin(y) ] + [ sin(x)cos(y)-cos(x)sin(y) ]
that messy expression above turns into
2sin(x)cos(y)
Notice how the cos(x)sin(y) terms go away. They add up to 0 and the 0 is "absorbed", more or less, into the other terms. This works since x+0 = 0+x = x for any real x.
There are two copies of the sin(x)cos(y) terms added up, so sin(x)cos(y)+sin(x)cos(y) = 2sin(x)cos(y)
So after adding those equations, we end up with sin(x+y)+sin(x-y) = 2sin(x)cos(y)
The last thing to do is multiply both sides by 1/2, then flip the equation and we end up with this:
sin(x)*cos(y) = 1/2 [ sin(x+y)+sin(x-y) ]
which is the first identity we wanted to prove.
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It's the same kind of story with the second equation given
cos(x)sin(y) = 1/2 [ sin(x+y)-sin(x-y) ]
Instead of adding sin(x+y) and sin(x-y), we're subtracting this time. Once again,
sin(x+y) = sin(x)cos(y)+cos(x)sin(y)
sin(x-y) = sin(x)cos(y)-cos(x)sin(y)
which leads to
sin(x+y)-sin(x-y) = 2cos(x)sin(y)
when we subtract the corresponding left and right hand sides together. The sin(x)cos(y) terms cancel this time.
Multiply both sides by 1/2 and rearrange the sides and we get
cos(x)sin(y) = 1/2 [ sin(x+y)-sin(x-y) ]
which proves the identity
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