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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 = θ
= x = tan θ, where `θ ∈ (- pi/2, pi/2)`
∴ sin (tan–1 x) = sin θ
Now, sin θ = `1/("cosec" θ)`
= `1/sqrt(1+cot^2θ)`
= `1/sqrt(1+ 1/tan^2θ)`
= `x/(sqrt(x^2 + 1)`
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