How do you simplify cos(x-y)-cos(x+y) to trigonometric functions of x and y?

1 Answer
Jan 15, 2016

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

Explanation:

We know

cos(alpha + beta) = cos(alpha)cos(beta) - sin(alpha)sin(beta)
cos(alpha - beta) = cos(alpha)cos(beta) + sin(alpha)sin(beta)

So from there we have

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