If t a n - 1 1 2 x + 1 + t a n - 1 1 4 x + 1 = c o t - 1 x 2 2 , then the number of all possible values of x is/are

Step-by-step solution

t a n - 1 1 2 x + 1 + 1 4 x + 1 1 - 1 2 x + 1 ⋅ 1 4 x + 1 = t a n - 1 2 x 2 t a n - 1 6 x + 2 8 x 2 + 6 x = t a n - 1 2 x 2 Therefore, 3 x + 1 4 x 2 + 3 x = 2 x 2 x 2 3 x + 1 = 2 4 x 2 + 3 x 3 x 3 - 7 x 2 - 6 x = 0 x 3 x 2 - 7 x - 6 = 0 x x - 3 3 x + 2 = 0 We take x = 3,0 because when x = - 2 3 , LHS of the given equation will be negative whereas t a n - 1 2 x 2 is positive. Hence, the number of values = 2