How to prove? $|\sin(nx)| \lt n|\sin(x)|$

Question

I want to know how to prove this?

For $n= 2,3,4,\ldots$ and $ 0<x<\frac{\pi}{2} $,
$$|\sin(nx)| \lt n|\sin(x)| $$

Answer 1

$$\frac{\sin Nx}{\sin x}=\frac{e^{Nix}-e^{-Nix}}{e^{ix}-e^{-ix}} =e^{-(N-1)ix}\frac{e^{2iNx}-1}{e^{2ix}-1} =e^{-(N-1)ix}(1+e^{2ix}+\cdots+e^{2(N-1)ix})$$ etc.