If sum of all the solutions of the equation 8 cos x · cos π 6 + x · cos π 6 - x - 1 2 = 1 in 0 , π is k π , then k is equal to:

Step-by-step solution

Given equation 8 cos x . cos π 6 + x . cos π 6 - x - 1 2 = 1 We would use the formula cos A + B . cos A - B = cos 2 A - sin 2 B Now, cos π 6 + x . cos π 6 - x = cos 2 π 6 - sin 2 x = 3 4 - sin 2 x ⇒ 8 cos x   3 4 - sin 2 x - 1 2 = 1 ⇒ 8 cos x   1 4 - sin 2 x = 1 ⇒ 8 cos x 1 4 - 1 - cos 2 x = 1 ⇒ 8 cos x cos 2 x - 3 4 = 1 ⇒ 8 cos 3 x - 6 cos x = 1 ⇒ 2 4 cos 3 x - 3 cos x = 1 ⇒ 2 cos 3 x = 1 ⇒ cos 3 x = 1 2 General Solution 3 x = 2 n &