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If $- 1 \leq x \leq 0,$ then $\cos^{- 1} (2 x^{2} - 1)$ equals

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a

2cos−1⁡x

b

π−2cos−1⁡x

c

2π−2cos−1⁡x

d

−2cos−1⁡x

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detailed solution

Correct option is C

Let cos−1⁡x=θ. Then, x=cos⁡θ.Also, −1≤x≤0⇒−1≤cos⁡θ≤0⇒π2≤θ≤πNow, cos−1⁡2x2−1=cos−1⁡(cos⁡2θ)=cos−1⁡(2π−2θ)⇒ cos−1⁡2x2−1=2π−2θ∵π2≤θ≤π⇒π≤2θ≤2π⇒0≤2π−2θ≤π⇒ cos−1⁡2x2−1=2π−2cos−1⁡x

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