Monotonicity of trigonometric function Consider the function
$$f(x)=\frac{\sin(x)}{\sin((2k+1)x)}$$
with $k$ a positive integer. It seems that $f$ is strictly increasing in $[0,\frac{\pi}{2(2k+1)}]$.  Is there some easy proof of this monotonicity property that does not invole differentiation (through suitable inequalities)?  

You have $\sin2x=2\sin x \cos x$ and $\sin 3x=\sin x \cos 2x + \cos x \sin 2x =\sin x (\cos 2 x + 2 \cos^2 x)$, and similar formulas for higher $\sin (kx)$. Then 
$$\frac{\sin(3x)}{\sin x} = \cos 2 x + 2\cos^2x
$$
where the RHS is decreasing in $x$ and positive as long as $\cos 2x>0$. That gives you an interval where $\frac{\sin x}{\sin 3x}$ is increasing.
Of course turning this into a proof for general $k$ is a lot more complicated than differentiating, but it should be possible. (You don't get non-integer $k$, though).
