Let S = x ∈ - π 2 , π 2 : 9 1 - tan 2 x + 9 tan 2 x = 10 and β = ∑ x ∈ S tan 2 x 3 , then 1 6 ( β - 14 ) 2 is equal to

Step-by-step solution

Given, 9 1 - tan 2 x + 9 tan 2 x = 10 Now let 9 tan 2 x = t , then above equation will be, 9 t + t = 10 ⇒ t 2 - 10 t + 9 = 0 ⇒ t = 9   or   t = 1 So, when 9 tan 2 x = 9 ⇒ tan 2 x = 1 ⇒ tan x = ± 1 ⇒ x = ± π 4 , as given x ∈ - π 2 , π 2 Now when 9 tan 2 x = 1 ⇒ tan 2 x = 0 ⇒ x = 0 Hence, β = ∑ x ∈ S x 3 = tan 2 0 3 + tan 2 π 12 + tan 2 - π 12 = 0 + 2 2 - 3 2 = 14 - 8 3 So, the value of 1 6 β - 14 2 = 64