IMO 1963

Let S= $\cos \dfrac\pi 7 + \cos\dfrac{3\pi}7 + \cos \dfrac{5\pi}7 \quad\quad \small{(\because - \cos\dfrac{2\pi}7= \cos\dfrac{5\pi}7)}$ $(2 \sin\dfrac{\pi}{7})\times S=\sin\dfrac{2\pi}7- \sin0+\sin\dfrac{4\pi}7- \sin\dfrac{2\pi}7+ \cos\dfrac{6\pi}7- \sin\dfrac{4\pi}7$ $=\sin\dfrac{6\pi}7 \small{\color{#20A900}{(\text{Using}\quad 2\cos A \sin B=sin(A-B)-sin(A+B)})}$ $\Rightarrow S=0.5 \dfrac{\sin\dfrac{6\pi}{7}}{\sin\dfrac{\pi}{7}}$ $=0.5$