If n is the number of solutions of the equation 2 cos x 4 sin π 4 + x sin π 4 - x - 1 = 1 , x ∈ 0 , π and S is the sum of all these solutions, then the ordered pair n , S is :

Step-by-step solution

2 cos x 4 sin π 4 + x sin π 4 - x - 1 = 1 ⇒     2 cos x 2 cos 2 x - 2 cos π 2 - 1 = 1 ⇒     2 cos x 2 . 2 cos 2 x - 1 - 1 = 1 ⇒   2 cos x 4 cos 2 x - 3 = 1 ⇒   4 cos 3 x - 3 cos x = 1 2 ⇒   cos 3 x = 1 2 ⇒   3 x = π 3 , 5 π 3 , 7 π 3 ⇒   x = π 9 , 5 π 9 , 7 π 9 No. of solutions = n = 3 Sum of solutions = S = π 9 + 5 π 9 + 7 π 9 = 13 π 9