how can I find the period of $f=\sin^4(x)$? How can I find the period of $f=\sin^4(x)$?
$\sin^4(x)=\frac{3-4\cos2x+\cos4x}{8}$. I didn't manage to reduce $\cos 4x$ to $\cos 2x$.

$\dfrac{3}{8}$ is constant, $-\dfrac{1}{2}\cos 2x$ has period $\dfrac{2\pi}{2} = \pi$, and $\dfrac{1}{8}\cos 4x$ has period $\dfrac{2\pi}{4} = \dfrac{\pi}{2}$. 
This should be enough to tell you that $\sin^4 x = \dfrac{3}{8} -\dfrac{1}{2}\cos 2x + \dfrac{1}{8}\cos 4x$ has period $\pi$. 
