Question

Prove that tan20 tan40 tan60 tan80 = 3

Answer 1

numerator
$\sqrt{3}/2$ (sin 20.sin40.sin80)
= $\sqrt{3}/4$ (2sin20.sin40). sin80
= $\sqrt{3}/4$ (cos20- cos60). sin80
=$\sqrt{3}/8$ (2sin80.cos20- sin80)
=$\sqrt{3}/8$ (sin100+ sin60 -sin80)
= $\sqrt{3}/8$ (sin80 -sin80+ $\sqrt{3}/2$)
=3/16

denominator
= 1/2 (cos20.cos40.cos80)
=1/4 (2cos20.cos40).cos80
=1/4 (cos60 +cos20).cos80
=1/8 (cos80+ cos20.cos80)
=1/8 (cos80 + cos100 +cos60)
=1/8 (cos80 -cos80 +1/2)
=1/16

=> numerator/denominator
=3

Answer 2

Answer:

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