Question

tan20 tan40 tan80 = root 3

Answer 1

tan 20° = 0.36
tan 40° = 0.84
tan 80° = 5.67

now,
L.H.S. is

= tan 20°.tan 40°.tan 80°
= 0.36*0.84*5.67
= 1.71
= √3
= R.H.S.

Answer 2

tan20 tan40 tan80=root3
first find
sin20.sin40.sin80
=1/2 {2sin20.sin40} sin80
=1/2 {cos20-cos60} sin80
=1/2sin80.cos20-1/2cos60sin80
=1/4 {2sin80.cos20}-1/4sin80
=1/4 (sin100+sin60}-1/4sin80
=1/4sin (180-80)+1/4sin60-1/4sin80
=1/4sin60=root3/8

in the same way
cos20.cos40.cos80 find
=1/2 {2cos20.cos40} cos80
=1/2 {cos60+cos20} cos80
=1/4cos80+1/4 {2cos20.cos80}
=1/4cos80+1/4 {cos100+cos60}
=1/8
now
sin20.sin40.sin80/cos20.cos40.cos80=tan20.tan40.tan80=(root3/8)/(1/8)=root3
hence proved