Question

Proof cos(x+y) cos(x-y)= cos2x + cos2y - 1 ?

Answer 1

`cos(x+y)cos(x-y)=1/2(cos2x+cos2y)`

not `cos2x+cos2y-1`

Explanation:

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

= `(cosxcosy-sinxsiny)(cosxcosy+sinxsiny)`

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

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

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

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

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

= `1/2(1+cos2x)-1/2(1-cos2y)`

= `1/2(cos2x+cos2y)`