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

1 Answer
Mar 13, 2017

cos(x+y)cos(x-y)=1/2(cos2x+cos2y)cos(x+y)cos(xy)=12(cos2x+cos2y)

not cos2x+cos2y-1cos2x+cos2y1

Explanation:

cos(x+y)cos(x-y)cos(x+y)cos(xy)

= (cosxcosy-sinxsiny)(cosxcosy+sinxsiny)(cosxcosysinxsiny)(cosxcosy+sinxsiny)

= cos^2xcos^2y-sin^2xsin^2ycos2xcos2ysin2xsin2y

= cos^2x(1-sin^2y)-(1-cos^2x)sin^2ycos2x(1sin2y)(1cos2x)sin2y

= cos^2x-cos^2xsin^2y-sin^2y+cos^2xsin^2ycos2xcos2xsin2ysin2y+cos2xsin2y

= cos^2x-sin^2ycos2xsin2y

= 1/2(2cos^2x)-1/2(2sin^2y)12(2cos2x)12(2sin2y)

= 1/2(1+cos2x)-1/2(1-cos2y)12(1+cos2x)12(1cos2y)

= 1/2(cos2x+cos2y)12(cos2x+cos2y)