16sin20.sin40.sin60.sin80=3 Prove.?

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Answer 1 · 2018-03-05T17:44:11

Explanation in below

$x = 16 sin 20 \cdot sin 40 \cdot sin 60 \cdot sin 80$ $x=16sin20*sin40*sin60*sin80$

= $16 \cdot \frac{\sqrt{3}}{2} \cdot sin 80 \cdot sin 40 \cdot sin 20$ $16*(sqrt3)/2*sin80*sin40*sin20$

= $8 \sqrt{3} \cdot sin 80 \cdot sin 40 \cdot sin 20$ $8sqrt3*sin80*sin40*sin20$

= $4 \sqrt{3} \cdot (cos 40 - cos 120) \cdot sin 20$ $4sqrt3*(cos40-cos120)*sin20$

= $4 \sqrt{3} \cdot (cos 40 + cos 60) \cdot sin 20$ $4sqrt3*(cos40+cos60)*sin20$

= $4 \sqrt{3} \cdot cos 40 \cdot sin 20 + 4 \sqrt{3} \cdot cos 60 - sin 20$ $4sqrt3*cos40*sin20+4sqrt3*cos60-sin20$

= $2 \sqrt{3} \cdot (sin 60 - sin 20) + 2 \sqrt{3} \cdot sin 20$ $2sqrt3*(sin60-sin20)+2sqrt3*sin20$

= $2 \sqrt{3} \cdot sin 60$ $2sqrt3*sin60$

= $2 \sqrt{3} \cdot \frac{\sqrt{3}}{2}$ $2sqrt3*(sqrt3)/2$

= $3$ $3$