<s>[INST] 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 $\cos 2x$.
 [/INST] Note that the equation $$\sin^4 (x+y)=\sin^4 x$$
Implies $$\sin^4 (x+y)-\sin^4 x = \left(\sin (x+y)-\sin x\right)\left(\sin (x+y)+\sin x\right)\left(\sin^2 (x+y)+\sin^2 x\right)=0$$
The final factor is nonnegative, and can only be zero when $\sin x=0$ so contributes no value of $y$ which works for all $x$.
So we have $$\sin (x+y)=\sin x\cos y+\cos x \sin y=\pm \sin x$$ And this has to be true for all $x$ and constant $y$. Set $x=0$ to obtain $\sin y=0$ and $y=n\pi$. So the period is $\pi$ (after checking that these values of $y$ work).
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