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$\cos^{- 1} \{\frac{1}{2} x^{2} + \sqrt{1 - x^{2}} \sqrt{1 - \frac{x^{2}}{4}}\} = \cos^{- 1} \frac{x}{2} - \cos^{- 1} x$ holds for

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a

x∈[−1, 1]

b

x∈R

c

x∈[0, 1]

d

x∈[−1, 0]

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detailed solution

Correct option is C

We observe thatx22+1−x21−x24 is positive and defined for all x∈[−1, 1]∴ cos−1⁡x22+1−x21−x24∈0,π2Now, RHS of the given relation is defined forx2≤1 and |x|≤1 i,e. for x∈[−1, 1]Also, LHS ≥0⇒ cos−1⁡x2−cos−1⁡x≥0⇒cos−1⁡x2≥cos−1⁡x⇒x∈[0,1]Hence, LHS = RHS for x∈[0, 1].

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