Class 12 Inverse Trigonometric Functions · Short Answer2 Marksinverse_trigonometric_identities
Class 12 Maths · Inverse Trigonometric Functions · Short Answer #58
Prove \(\sin^{-1}(2x\sqrt{1-x^2})=2\cos^{-1}x\) for \(1/\sqrt2\le x\le1\).
✅ Step-by-Step Solution
Final Answer: Identity proved on [1/√2,1].
- Let θ=arccos x. The domain gives 0≤θ≤π/4 and √(1−x²)=sin θ.
- Then 2x√(1−x²)=2cos θ sin θ=sin2θ, with 2θ∈[0,π/2].
- Therefore arcsin(2x√(1−x²))=2θ=2arccos x, as required.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Let θ=arccos x. The domain gives 0≤θ≤π/4 and √(1−x²)=sin θ.
- [1 mark] Then 2x√(1−x²)=2cos θ sin θ=sin2θ, with 2θ∈[0,π/2].
- [0.5 marks] Therefore arcsin(2x√(1−x²))=2θ=2arccos x, as required.
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