I am trying to prove this inequality by induction:
For all $x$ in the interval $x\in [0, \pi]$, prove that: $$ |\sin (nx)| \leq n\sin(x) \textit{, n a nonnegative integer}$$
The base case is pretty trivial so after I assumed that this inequality holds for $k$ I want to show that it holds for $P(k+1)$ then
$$ |\sin[(k+1)x]| \leq (k+1)\sin x = k\sin x + \sin x $$
I am not really sure how to proceed from here, or if my method is not very good. Anyways any help would be really great, thanks!
Hint: We have $\sin(a+b)=\sin a\cos b+\cos a\sin b$. Let $a=kx$ and $b=x$.