<s>[INST] Comparing the expansions of $\sin ^3 (x)$ and $\cos ^3 (x)$ By using $(\cos x+i\sin x)^3=\cos 3x+ i \sin 3x$,
We have:
$\sin ^3x= \frac{3}{4} \sin x- \frac{1}{4} \sin 3x$
$\cos ^3 x=\frac{3}{4}\cos x + \frac{1}{4}\cos 3x$
If we compare the coefficients of the equations we see coefficient of $\sin x$ and $\cos x$ are both $\frac{3}{4}$ and coefficient of $\sin 3x $  and $\cos 3x$ are $-\frac{1}{4}$ and $\frac{1}{4}$ respectively. the coefficients are very similar to each other !
Is there any approach to make me understand by intuition that why the coefficients are so similar to each other in these two equations?
 [/INST] Simply do the substitution $x \mapsto x + \frac \pi2$: Then $\sin x$ becomes $\sin (x + \frac\pi2)$, which is $\cos x$. And $\sin 3x$ becomes $\sin (3x + \frac{3\pi}2)$, which is $-\cos 3x$.
Note that similar substitutions do not exist for $\sin 2x$ and $\cos 2x$, so there's nothing similar for $\sin 2x$. That's where I got the false impression that you can't express $\sin 3x$ as a polynomial of $\sin x$.
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