Consider the equation l o g 2 s i n x 1 + c o s x = 2 , x ∈ - π 2 , 3 π 2 . If the sum of the roots is p π q , where G.C.D p , q = 1 , then the value of p 2 + q 2 is

Step-by-step solution

l o g 2 s i n x 1 + c o s x = 2 ….(i) 2 s i n x ≠ 1 , 2 s i n x > 0 , 1 + c o s x > 0 ⇒ s i n x ≠ 1 2 , s i n x > 0 and x ≠ 0 or a multiple of π ⇒ x ∈ 0 , π - π 4 , 3 π 4 (feasible region) Hence, from (i), we get, 2 s i n x 2 = 1 + c o s x ⇒ 2 s i n 2 x = 1 + c o s x ⇒ 2 c o s 2 x + c o s x - 1 = 0 ⇒ 2 c o s x - 1 c o s x + 1 = 0 ⇒ c o s x = 1 2 … c o s x + 1 > 0 ⇒ x = π 3 ⇒ p = 1 , q = 3 ⇒ p 2 + q 2 = 10