lean_workbook_plus_59884

Given the identity $a^2 + b^2 = (a + b)^2 - 2ab$ and $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$, prove that $\sin^6{x} + \cos^6{x} - 1 = -3\sin^2{x}\cos^2{x}$ using these identities.