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I have to show that for every $x \in \mathbb{R}$ and every $n \in \mathbb{N}$
$$|\sin(nx)| \le n|\sin(x)| $$ In the previous exercise, I have showed that $|\sin(x)|≤|x|$ with the use of the mean value theorem. I think that I cannot use this approach this time. I also tried to write $\sin(nx)$ as a series expansion but that doesn't work either.

Does anyone know how I can solve this?

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It is easy to see by induction. Keep the fact that, $|\cos x|\leq 1$. For $n=2$ $$|\sin(2x)| = |2\sin x \cos x|\leq 2|\sin x |$$

Assuming $|\sin(nx)| \leq n|\sin x|$ and using the addition formula we have

$$|\sin((n+1)x)| = |\sin nx ~\cos x +\sin x~ \cos nx|\\\overset{|\cos(\cdot)|\leq 1}{\leq}|\sin nx| + |\sin x| \overset{\text{by assumption}}{\leq}(n+1) |\sin x| .$$

For your worry

$$ |\sin(nx)| = |\int_0^{nx} \cos t\, dt\leq |nx|.$$

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You can use what you have already proved.

Apply it for "$x=nu$", then you get

$$\vert \sin(nu)\vert\leqslant \vert nu\vert=n\vert u\vert$$

since $n\geqslant 0$.


Edit.

For your new question, you should prove it by induction using the fact that

$$\vert a+b\vert \leqslant \vert a\vert+\vert b\vert.$$

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$$ |\sin(nx)| = |\int_0^{nx} \cos t\, dt\leq |nx|$$

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