Question 12
(GRE) Consider on the first quadrant of the trigonometric circle the arcs $AM_1=x_1$, $AM_2=x_2$, $AM_3=x_3$, $\ldots$, $AM_\nu=x_\nu$, such that $x_1\lt x_2\lt x_3\lt \cdots\lt x_\nu$. Prove that
$$\sum_{i=0}^{\nu-1}\sin2x_i-\sum_{i=0}^{\nu-1}\sin(x_i-x_{i+1})\lt \frac\pi2+\sum_{i=0}^{\nu-1}\sin(x_i+x_{i+1}).$$
$$\sum_{i=0}^{\nu-1}\sin2x_i-\sum_{i=0}^{\nu-1}\sin(x_i-x_{i+1})\lt \frac\pi2+\sum_{i=0}^{\nu-1}\sin(x_i+x_{i+1}).$$