Question

tan20°tan40°tan80°=√3
​

Answer 1

Given :

• tan20°tan40°tan80°

To prove :

• tan20°tan40°tan80° =.√3

Solution :

We have,

LHS = tan 20° tan 40° tan 80°

= sin 20°.sin 40°.sin 80°/cos 20°.cos 40°.cos80°

= (2sin 20°. sin 40° ) sin80°/(2cos20°.cos40°) cos 80°

= (cos 20°-cos 60°) sin 80°/(cos 60° + cos 20°)cos 80°

= sin 80°.cos 20° ½sin 80°/½ cos80° + cos80° .cos20°

= 2 sin 80°.cos 20°- sin 80°/cos80° + 2cos 80° cos 20°

= sin100° + sin 60°- sin 80°/cos 80° + cos100°+ cos 60°

= sin(180°- 80°) + sin 60 - sin 80/cos 80°+cos(180°- 80°) + cos 60°

= sin 80 + sin 60°- sin 80°/cos80°- cos80°+ cos60°

= sin 60°/cos 60°

= tan 60°

= √3 ........RHS