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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))`
Solution
sin (tan–1 x), | x| < 1 is equal to `underline (x/(sqrt(1+ x^2)))`.
Explanation:
Let tan-1 x = θ
= x = tan θ, where θ ∈ `(- pi/2, pi/2)`
∴ `sin (tan^-1x) = sin theta`
Now,
`= sin theta = 1/(cosectheta) = 1/sqrt(1+cot^2theta)`
= `1/sqrt(1+ 1/tan^2theta)`
= `x/(sqrt(x^2 + 1)`
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