cos(x+y)cos(x-y)cos(x+y)cos(x−y)
= (cosxcosy-sinxsiny)(cosxcosy+sinxsiny)(cosxcosy−sinxsiny)(cosxcosy+sinxsiny)
= cos^2xcos^2y-sin^2xsin^2ycos2xcos2y−sin2xsin2y
= cos^2x(1-sin^2y)-(1-cos^2x)sin^2ycos2x(1−sin2y)−(1−cos2x)sin2y
= cos^2x-cos^2xsin^2y-sin^2y+cos^2xsin^2ycos2x−cos2xsin2y−sin2y+cos2xsin2y
= cos^2x-sin^2ycos2x−sin2y
= 1/2(2cos^2x)-1/2(2sin^2y)12(2cos2x)−12(2sin2y)
= 1/2(1+cos2x)-1/2(1-cos2y)12(1+cos2x)−12(1−cos2y)
= 1/2(cos2x+cos2y)12(cos2x+cos2y)