If ∫ 0 π 2 c o t x c o t x + cosec x d x = m ( π + n ) , then m n is equal to

Step-by-step solution

Let I = ∫ 0 π 2 cot x cot x + cosec x d x ⇒ I = ∫ 0 π 2 c o s x c o s x + 1 d x ⇒ I = ∫ 0 π 2 c o s x + 1 - 1 c o s x + 1 d x ⇒ I = ∫ 0 π 2 1 - 1 2 cos 2 x 2 d x ⇒ I = ∫ 0 π 2 1 - 1 2 s e c 2 x 2 d x ⇒ I = x - tan x 2 0 π 2 ⇒ I = π 2 - 1 - 0 - 0 ⇒ I = π 2 - 1 = 1 2 π - 2 . Now using given information ∫ 0 π 2 cot x cot x + cosec x d x = m ( π + n ) , clearly m = 1 2 and n = - 2 . &#865