If the solution of the equation log cos x cot x + 4 log sin x tan x = 1 , x ∈ 0 , π 2 is sin - 1 α + β 2 , where α , β are integers, then α + β is equal to:

Step-by-step solution

Given: log cos x cot x + 4 log sin x tan x = 1 ,   x ∈ 0 , π 2 ⇒ lncos x - lnsin x lncos x + 4 lnsin x - lncos x lnsin x = 1 ⇒ 1 - lnsin x lncos x + 4 1 - lncos x lnsin x = 1 ⇒ lnsin x lncos x + 4 lncos x lnsin x - 4 = 0 ⇒ ( lnsin x ) 2 - 4 ( lnsin x ) ( lncos x ) + 4 ( lncos x ) 2 = 0 ⇒ lnsin x - 2 lncos x 2 = 0 ⇒ lnsin x = 2 lncos x ⇒ lnsin x = lncos 2 x ⇒ cos 2 x = sin x ⇒ 1 - sin 2 x = sin x ⇒ sin 2 x + sin x - 1 = 0 ⇒ sin x = - 1