Prove that the inequality $\sin^8(x) + \cos^8(x) \geq \frac{1}{8}$ is true for every real number. Prove that the inequality $\sin^8(x) + \cos^8(x) \geq \frac{1}{8}$ is true for every real number.

By derivation, the extrema are reached for
$$\cos(x)\sin^7(x)-\sin(x)\cos^7(x)=0=\cos(x)\sin(x)(\sin^6(x)-\cos^6(x)).$$
They occur at multiples of $\frac\pi4$, where the function equals $0+1$ or $\frac1{16}+\frac1{16}$ or $1+0$.
