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If $2 \tan^{- 1} x + \sin^{- 1} (\frac{2 x}{1 + x^{2}})$ is independent of $x,$ then

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a

x∈[1,∞)∈(−∞,−1)

b

x∈[−1,1]

c

x∈(-∞,1]

d

none of these

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detailed solution

Correct option is A

We know that sin−1⁡2x1+x2=2tan−1⁡x, if −1≤x≤1π−2tan−1⁡x, if x≥1−π−2tan−1⁡x, if x≤−1∴ 2tan−1⁡x+sin−1⁡2x1+x2=4tan−1⁡x, if −1≤x≤1π, if x≥1−π, if x≤−1Thus, 2tan−1⁡x+sin−1⁡2x1+x2 is independent of x, if x∈[1,∞) or, x∈(−∞,−1].

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