Question

Prove that: tan10tan50tan70=tan30

Answer 1

tan 10  tan 50  tan 70 
=tan 10 tan(60-10) tan(60+10)
= tan10*(tan60- tan10)/(1+tan60 tan10) * (tan60+tan10)/(1-tan60tan10)
= tan10*(tan^2 60 - tan^2 10)/(1 - tan^2 60 tan^2 10) 
= tan10 * (3 - tan^2 10)/(1 - 3 tan^2 10)
= [3 tan10 - tan^3 10 ] /( 1 - 3 tan^2 10)
= tan (3 * 10)   using the formula tan 3A = (3 tanA-tan^3A)/(1-3tan^2 A)
= tan 30