Given that the inverse trigonometric functions take principal values only. Then, the number of real values of x which satisfy sin - 1 3 x 5 + sin - 1 4 x 5 = sin - 1 x is equal to:

Step-by-step solution

sin - 1 3 x 5 + sin - 1 4 x 5 = sin - 1 x sin - 1 3 x 5 1 - 16 x 2 25 + 4 x 5 1 - 9 x 2 25 = sin - 1 x 3 x 5 1 - 16 x 2 25 + 4 x 5 1 - 9 x 2 25 = x x = 0 ,   3 25 - 16 x 2 + 4 25 - 9 x 2 = 25 4 25 - 9 x 2 = 25 - 3 25 - 16 x 2 squaring we get 16 25 - 9 x 2 = 625 + 9 25 - 16 x 2 - 150 25 - 16 x 2 400 = 625 + 225 - 150 25 - 16 x 2 25 - 16 x 2 = 3 ⇒ 25 - 16 x 2 = 9 ⇒ x 2 = 1 Put x = 0 , 1 , - 1 in the original equation We see that all values satisfy the original equation. Number of solution = 3