cos^2(2x) - sin^2(x)

= (cos^2(x) - sin^2(x))^2 - sin^2(x)

= (cos^2(x) - 1 + cos^2(x))^2 - 1 + cos^2(x)

= (2 cos^2(x) - 1)^2 - 1 + cos^2(x)

= 4 cos^4(x) - 4 cos^2(x) + 1 - 1 + cos^2(x)

= 4 cos^4(x) - 3 cos^2(x)

= cos^2(x)(4 cos^2(x) - 3)

= cos^2(x)(2cos(x) - sqrt(3))(2cos(x) + sqrt(3))