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Question
sin (tan–1 x), |x| < 1 is equal to ______.
Options
`x/(sqrt(1 - x^2))`
`1/sqrt(1 - x^2)`
`1/sqrt(1 + x^2)`
`x/(sqrt(1 + x^2))`
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Solution
sin (tan–1 x), |x| < 1 is equal to `bbunderline (x/(sqrt(1+ x^2)))`.
Explanation:
Let tan–1 x = y.
Then, tan y = x
⇒ `sin y = x/sqrt(1 + x^2)`
∴ `y = sin^-1 (x/sqrt(1 + x^2))`
⇒ `tan^-1x = sin^-1 (x/sqrt(1 + x^2))`
∴ `sin (tan^-1x) = sin (sin^-1 x/sqrt(1 + x^2))`
= `x/sqrt(1 + x^2)`
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