The number of solutions of the equation sin - 1 x 2 + 1 3 + cos - 1 x 2 - 2 3 = x 2 for x ∈ [ - 1 , 1 ] , and [ x ] denotes the greatest integer less than or equal to x , is :

Step-by-step solution

Given equation is sin - 1 x 2 + 1 3 + cos - 1 x 2 - 2 3 = x 2 Now, sin - 1 x 2 + 1 3 is defined if - 1 ≤ x 2 + 1 3 ≤ 1 ⇒ - 1 ≤ x 2 + 1 3 < 2 ⇒ - 4 3 ≤ x 2 < 5 3 ⇒ 0 ≤ x 2 < 5 3       . . . 1 Also, and cos - 1 x 2 - 2 3 is defined if - 1 ≤ x 2 - 2 3 ≤ 1 ⇒ - 1 ≤ x 2 - 2 3 < 2 ⇒ - 1 3 ≤ x 2 < 8 3 ⇒ 0 ≤ x 2 < 8 3       . . . 2 So, from ( 1 ) and ( 2 ) we can conclude 0 &#