Question

Question #588d8

Answer 1

Use the identities:

`sin(x+y)=sin(x)cos(y)+cos(x)sin(y)`
`cos(x-y)=cos(x)cos(y)+sin(x)sin(y)`
`sin^2x+cos^2x=1`

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We will start with the LHS:

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

`(sinxcosy+cosxsiny)(cosxcosy+sinxsiny)`

Now use the FOIL method (from the good ol' days back in Algebra 1)

`sinxcosxcos^2y+sin^2xsinycosy+cos^2xsinycosy+sinxcosxsin^2y`

`(sin^2y+cos^2y)(sinxcosx)+(sin^2x+cos^2x)(sinycosy)`

Now use the third identity listed above.

`sinxcosx+sinycosy`

In fact, `sin(x+y)cos(x-y)` does NOT always equal `sin^2x-sin^2y`. It DOES however equal `1/2(sin2x+sin2y)`, if that was somehow what was meant to be written.