Question

$\cos \frac{\pi }{5}\cos \frac{2\pi }{5}=\frac{1}{4}andcos\frac{\pi }{5}+\cos \frac{3\pi }{5}=\frac{1}{2}$ without using standard value

Answer 1

$L.H.S=\frac{1}{2^{2}sin\left(\pi /5\right)}\cdot [sin{2}^2+\frac{\pi }{5}]$
$L.H.S=\frac{1}{4sin\left(\pi /5\right)}\cdot sin[\pi -\frac{\pi }{5}]=\frac{1}{4}$ ....(1)
Also $cos\frac{\pi }{5}+cos\frac{3\pi }{5}=2\cdot cos\frac{\pi }{5}cos\frac{\pi }{5}$
$=2\cdot \frac{1}{4}=\frac{1}{2}$...by (1)