Question

Prove that tan20tan30tan40tan80 = 1​

Answer 1

the answer is simple.... prove that cos20 * cos40 * cos60 * cos80 = 1/16 and sin20 * sin40*sin60 * sin80 = 3/16.... then simply put these results in this questions.(the above proofs can be found on the internet, even on brainly)

so, LHS = tan30 * (tan20 *  tan40 * tan80)

=> 1/$\sqrt[]{3}$ * (sin20 * sin40 * sin80)/(cos20 * cos40 * cos80)

=> 1/$\sqrt[]{3}$ * (3/16 sin 60)/(1/16 cos 60)

=> 1/$\sqrt[]{3}$ * (3 * 1/2 * 16)/(1 * $\sqrt[]{3}$/2 * 16)

=> 1/$\sqrt[]{3}$ * (3/2)/($\sqrt[]{3}$/2) = 3/2 * 2/3 = 1 = RHS