Question

If cos2x=m, then sin^6 x + cos^6 x=?

Answer 1

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)