cos2x = m.
Now 1 = cos^2x + sin^2x, so,
(1)^3 = cos^6x + 3cos^4x.sin^2x + 3cos^2x.sin^4x + sin^6x
1 = (cos^6x + sin^6x) + 3cos^2x.sin^2x(cos^2x + sin^2x)
1 = (cos^6x + sin^6x) + (3/4)(sin2x)^2
1 = (cos^6x + sin^6x) + (3/4)(1 - cos^2(2x))
1 = (cos^6x + sin^6x) + (3/4)(1 - m^2)
cos^6x + sin^6x = 1 - (3/4)(1 - m^2)
cos^6x + sin^6x = (1/4)(1 + 3m^2)