trigonometry - How to prove by induction that $|sin(nx)| leq n|sin x|$?

Here $n$ belongs to natural numbers. Firstly, I proved the relation by putting $n = 1$ . Then, taking

$$|\sin(mx)| \leq m|\sin x|$$ true, I had to prove $$|\sin(m + 1)x| \leq (m + 1)|\sin x|$$ Now, here I got stuck. How to prove it?? Please help.

Answer

The inductive step: Using the triangle inequality and the fact that $\sin$ and $\cos$ function are bounded by $1$ we get

$$|\sin((n+1)x)|=|\sin(nx)\cos x+\cos(nx)\sin(x)|\le|\sin(nx)|cos(x)|+|\cos(nx)||sin(x)|\\\le|\sin(nx)|+|sin(x)|\le n|\sin(x)|+|\sin(x)|=(n+1)|\sin(x)|$$