<s>[INST] How can I find the following product? $ \tan 20^\circ \cdot \tan 40^\circ \cdot \tan 80^\circ.$ How can I find the following product using elementary trigonometry?
$$ \tan 20^\circ \cdot \tan 40^\circ \cdot \tan 80^\circ.$$
I have tried using a substitution, but nothing has worked.
 [/INST] tan 20 tan 40 tan 80 = (sin 20 sin 40 sin 80) / (cos 20 cos 40 cos 80) 

Solving for numerator : 

sin 20 sin 40 sin 80 = (sin 80 sin 20)(sin 60 sin 40) / (sin 60) 
= (1/2)(cos 60 - cos 100)(1/2)(cos 20 - cos 100) / (sin 60) ... (product rule) 
= (1/4)(1/2 + sin 10)(cos 20 + sin10) / (sin 60) ... (cos 60 = 1/2 & cos 100 = -sin 10) 
= [(1/8)cos 20 + (1/8)sin 10 + (1/4)sin 10 cos 20 + (1/4) sin^2 10] / (sin 60) 
= [(1/8)cos 20 + (1/8)sin 10 + (1/8)(sin -10 + sin 30) + (1/4)(1/2)(1 - cos 20)] / (sin 60) 
= [(1/8(1/2) + 1/8] / (√3/2) ... (sin 30 = 1/2) 
= √3/8 

Solving for denominator : 

sin 160 
= 2sin 80 cos 80 
= 4sin 40 cos 40 cos 80 
= 8 sin 20 cos 20 cos 40 cos 80 
But, sin 160 = sin 20 
So, sin 20 = 8 sin 20 cos 20 cos 40 cos 80 
Cancel sin 20. Therefore, cos 20 cos 40 cos 80 = 1/8 

So, tan 20 tan 40 tan 80 = numerator/denominator = (√3/8) / (1/8) = √3 = tan 60 
Thus, x = 60º. (This is just the principal value, being one of an infinite other values) `enter preformatted text here`

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