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If $\sin^{- 1} \sqrt{x^{2} + 2 x + 1} + \sec^{- 1} \sqrt{x^{2} + 2 x + 1} = \frac{π}{2}, x \neq 0$ then the value of $2 \sec^{- 1} \frac{x}{2} + \sin^{- 1} \frac{x}{2}$ is equal to

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a

−π2 only

b

−3π2, π2

c

3π2 only

d

−3π2 only

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detailed solution

Correct option is C

We have,sin−1⁡x2+2x+1+sec−1⁡x2+2x+1=π2sin−1⁡x2+2x+1+cos−1⁡1x2+2x+1=π2⇒x2+2x+1=1x2+2x+1⇒x2+2x+1=1⇒x=0,−2.For x=0, we find that sec−1⁡x2 is not defined.For x=-2, we have2sec−1⁡x2+sin−1⁡x2=2sec−1⁡(−1)+sin−1⁡(−1)=2×π−π2=3π2

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