Question

If `sin^-1 ((2"a")/(1 + "a"^2)) + cos^-1 ((1 - "a"^2)/(1 + "a"^2)) = tan^-1 ((2x)/(1 - x^2))`. where a, x ∈ ] 0, 1, then the value of x is ______.

Options

• 0

• `"a"/2`

• a

• `(2"a")/(1 - "a"^2)`

Answer 1

If `sin^-1 ((2"a")/(1 + "a"^2)) + cos^-1 ((1 - "a"^2)/(1 + "a"^2)) = tan^-1 ((2x)/(1 - x^2))`. where a, x ∈ ] 0, 1, then the value of x is `(2"a")/(1 - "a"^2)`.

Explanation:

We have, `sin^-1   (2"a")/(1 + "a"^2) + cos^-1  (1 - "a"^2)/(1 + "a"^2) = tan^-1  (2x)/(1 - x^2)`

⇒ `2tan^-1"a" + 2tan^-1"a" = 2tan^-1x` .....`[(because 2tan^-1x = tan^-1  (2x)/(1 - x^2)),(2tan^-1x = sin^-1  (2x)/(1 + x^2)),(2tan^-1x = cos^-1  (1 - x^2)/(1 + x^2))]`

⇒ `2tan^-1"a" = tan^-1x`

⇒ `tan^-1  (2"a")/(1 - "a"^2) = tan^-1x`

⇒ x = `(2"a")/(1 - "a"^2)`