Question

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

Answer 1

`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)`