Question

How do you prove `sin(x+y)*sin(x-y)=(sinx)^2-(siny)^2`?

Answer 1

Using the angle sum and difference formula of the function sinus on the first member:

`(sinxcosy+cosxsiny)(sinxcosy-cosxsiny)=`

`sin^2xcos^2y-cos^2xsin^2y=`

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

`=sin^2x-sin^2xsin^2y-sin^2y+sin^2xsin^2y=`

`=sin^2x-sin^2y`

that is the second member.