MHB Can you prove this inequality involving trigonometric functions?

[Chris L T521]

The following was proposed by CaptainBlack.

Problem: Prove that $|\sin(nx)|\leq n|\sin(x)|$ for all $x\in\mathbb{R}$ and $n\in\mathbb{N}_+$.

There are no hints for this problem (Smile)

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[Jameson]

Congratulations to the following members for their correct solutions:

1) Amer
2) Sudharaka

Spoiler

The base case $n=1$ holds obviously and trivially.

Now suppose that for some $k \in \mathbb{N}_+$:
\[|\sin(kx)| \le k |\sin(x)|\].

Now consider:
\[|\sin((k+1)x)|=| \sin(kx)\cos(x)+\cos(kx)\sin(x)| \].

Then by the triangle inequality we get:
\[|\sin((k+1)x)|\le |\sin(kx)\cos(x)|+|\cos(kx)\sin(x)| \\ \phantom{[ \sin((k+1)x)xxx}\le |\sin(kx)| + |\sin(x)|=(k+1)|\sin(x)|\].

QED

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