2011/071) Show that 2 tan 20˚ + 4 tan 40˚ + 8 tan 80˚ = 9 (cot 10˚ - tan 10˚)

we first prove one relationship
cot x - tan x = cos x/ sin x - sin x/ cos x = (cos^2 x - sin ^2 x) /( sinx cos x) = 2 cot 2x

so

cot x - tan x = 2 cot 2x ...1

also
tan x = cot x - 2 cot 2x ...2

from 2 we generate following 3 relationships

2 tan 20= 2 cot 20 - 4 cot 40 ... 3
4 tan 40 = 4 cot 40 - 8 cot 40... 4
8 tan 80 = 8 cot 80 - 16 cot 160... 5
adding we get
2 tan 20˚ + 4 tan 40˚ + 8 tan 80 = 2 cot 20 - 16 cot 160
= 2 cot 20 + 16 cot(180-160)as cot x = - cot(180-x)
= 2 cot 20 + 16 cot 20
= 18 cot 20
= 9( 2 cot 20)
= 9( cot 10 - tan 10)from 1
proved