AMC 12 Mock Exam 5 Problem 11

Let $s=\sin^2x$ and $c=\cos^2x$, so $s+c=1$. Then $s^3+c^3=(s+c)^3-3sc(s+c)=1-3sc=1-3\sin^2x\cos^2x=1-\tfrac34\sin^22x$. Since $\sin^22x\le1$, with equality at $x=\frac\pi4$, the least value is $1-\frac34=\boxed{\tfrac14}$.