I'm having trouble making sense of addition and subtraction in trig identities.
cos^6x + sin^6x = 1 - 3 sin^2x + 3 sin^4x
Notice that $\displaystyle \cos^6{x} = (\cos^2{x})^3$
$\displaystyle = (1 - \sin^2{x})^3$
$\displaystyle = \sum_{k = 0}^{3}\left(_k^n\right)1^{n - k}+(-\sin^2{x})^k$
$\displaystyle = 1 - 3\sin^2{x} + 3\sin^4{x} - \sin^6{x}$.
So we can rewrite the original function as
$\displaystyle 1 - 3\sin^2{x} + 3\sin^4{x} - \sin^6{x} + \sin^6{x}$.
What does this equal?