Prove that, 8 sin20° sin40° sin80° = $\sqrt{3}$
Solution:
8 sin20° sin40° sin80°
= 4(2 sin20° sin40°) sin80°
=4 {cos(20°+40°)-cos(20°-40°)} sin80°
=4{cos60° –cos(-20°)} sin80°
= 4 (cos60° –cos20°) sin80°
= 4 $\left(\frac{1}{2} – cos20°\right)$ sin80°
=4×$\frac{1}{2}$ sin80° – 4 cos20° sin80°
= 2sin80° – 2(2 cos20° sin80°)
=2sin80°-2 {sin(20°+80°)+sin(80°-20°)}
= 2 sin80°-2(sin100° -sin60°)
= 2 sin80° -2sin100°+ 2sin60°
= 2 sin80° – 2sin100°+ 2× $\frac{\sqrt{3}}{2}$
= 2sin80° – 2sin100° + $\sqrt{3}$
= 2sin80° – 2sin(180°-80°) + $\sqrt{3}$
= 2sin80° – 2 sin80° + $\sqrt{3}$
= $\sqrt{3}$ [Proved]