Considering the principal values of the inverse trigonometric functions, the sum of all the solutions of the equation cos - 1 x - 2 sin - 1 x = cos - 1 2 x is equal to

Step-by-step solution

Given, cos - 1 x - 2 sin - 1 x = cos - 1 2 x ⇒ cos - 1 x - 2 π 2 - cos - 1 x = cos - 1 2 x ⇒ cos - 1 x - π + 2 cos - 1 x = cos - 1 2 x ⇒ 3 cos - 1 x = π + cos - 1 2 x           . . . 1 Taking cos both side we get, ⇒ cos 3 cos - 1 x = cos π + cos - 1 2 x ⇒ cos cos - 1 4 x 3 - 3 x = - cos cos - 1 2 x ⇒ 4 x 3 - 3 x = - 2 x ⇒ 4 x 3 = x ⇒ x = 0 , ± 1 2 All will satisfy the original equation, So, sum of all the solution will