The problem is that:
Show that for all $a \in \mathbb{R}$, $$|\sin(ax)\sin\left((a+\tfrac12)x\right)|\geq |\sin^2(ax)\cos(\tfrac{x}{2})|-|\sin(\tfrac{x}{2})|.$$
My idea is through simple calculation to show that $\sin(ax)\sin\left((a+\tfrac{1}{2})x\right) = \sin^2(ax)\cos(\tfrac{x}{2}) - \sin(\tfrac{x}{2})$ which I failed at, $\sin(ax)\sin\left((a+\frac{1}{2})x\right) = \cos(\frac{x}{2})(1-\sin^2(ax))-\cos(ax)\sin(ax)\sin(\frac{x}{2}).$ The hint from the problem(a) is that $\frac{1}{2} + \sum_{n=1}^{N}\cos(nx)= \pi D_N(x)$, and the general formula for $D_N(x)$, but I don't know how to use it. Thanks!