If tan π 9 , x , tan 7 π 18 are in arithmetic progression and tan π 9 , y , tan 5 π 18 are also in arithmetic progression, then x - 2 y is equal to :

Step-by-step solution

x = 1 2 tan π 9 + tan 7 π 18 and 2 y = tan π 9 + tan 5 π 18 so, x - 2 y = 1 2 tan π 9 + tan 7 π 18 - tan π 9 + tan 5 π 18 ⇒ x - 2 y = cot π 9 - tan π 9 2 - tan 5 π 18 ⇒ x - 2 y = cot 2 π 9 - 1 2 c o t π 9 - tan 5 π 18 ⇒ x - 2 y = cot 2 π 9 - tan 5 π 18 = 0 ∵ tan 5 π 18 = c o t π 2 - 5 π 18 = cot 2 π 9 ∵ tan 7 π 18 = c o t π 2 - 7 π 18 = cot π 9 ∵ cot 2 θ -