If m and n respectively are the numbers of positive and negative value of θ in the interval - π , π that satisfy the equation cos 2 θ cos θ 2 = cos 3 θ cos 9 θ 2 , then m n is equal to _____ .

Step-by-step solution

We have, cos 2 θ · cos θ 2 = cos 3 θ · cos 9 θ 2 ⇒ 2 cos 2 θ cos θ 2 = 2 cos 9 θ 2 cos 3 θ ⇒   cos 5 θ 2 + cos 3 θ 2 = cos 15 θ 2 + cos 3 θ 2 ⇒ cos 15 θ 2 = cos 5 θ 2 ⇒ 15 θ 2 = 2 k π ± 5 θ 2 ;   k ∈ Z ⇒ 5 θ = 2 k π or 10 θ = 2 k π ⇒ θ = 2 k π 5   or   θ = k π 5 ∴ θ = - π , - 4 π 5 , - 3 π 5