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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)`
Solution
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)`
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