Question from http://www.sgforums.com/forums/2297/topics/330176
Prove that tan x = cot x - 2 cot 2x and hence show that
2 tan 20 + 4 tan 40 + 8 tan 80 = 9(cot 10 - tan 10)
* all units are in degree form.
Answer:
RHS = cot x - 2 cot 2x
= cos x / sin x - 2 cos 2x /sin 2x
= cos x / sin x - 2 ([cos x]2 - [sin x]2) / 2 sin x cos x
= ( [cos x]2 - [cos x]2 + [sin x]2) / sin x cos x
= [sin x]2 / sin x cos x
= sin x / cos x
= tan x = LHS (proved)
Using tan x = cot x - 2 cot 2x
LHS = 2 tan 20 + 4 tan 40 + 8 tan 80
= 2 cot 20 - 4 cot 40 + 4 cot 40 - 8 cot 80 + 8 cot 80 - 16 cot 160
= 2 cot 20 - 16 cot 160
Note that tan 160 = -tan 20 since tan (180 - x) = -tan x
so 16 cot 160 = -16 cot 20
2 cot 20 - 16 cot 160
= 2 cot 20 + 16 cot 20
= 18 cot 20
= 9 (2 cot 20)
but, rearranging equation, 2 cot 2x = cot x - tan x
Hence, 9 (2 cot 20) = 9 (cot 10 - tan 10)
=> 2 tan 20 + 4 tan 40 + 8 tan 80 = 9 (cot 10 - tan 10)
