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Class 12 Inverse Trigonometric Functions · Short Answer2 Marksinverse_trigonometric_identities

Class 12 Maths · Inverse Trigonometric Functions · Short Answer #89

Prove \(\sin^{-1}(2x\sqrt{1-x^2})=2\sin^{-1}x\) for \(-1/\sqrt2\le x\le1/\sqrt2\).

✅ Step-by-Step Solution

Final Answer: Identity proved.
  1. Put θ=arcsin x, so θ∈[−π/4,π/4] and cos θ=√(1−x²)≥0.
  2. Then 2x√(1−x²)=sin2θ, with 2θ∈[−π/2,π/2], the arcsine principal interval.
  3. Therefore arcsin(2x√(1−x²))=2θ=2arcsin x.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Put θ=arcsin x, so θ∈[−π/4,π/4] and cos θ=√(1−x²)≥0.
  • [1 mark] Then 2x√(1−x²)=sin2θ, with 2θ∈[−π/2,π/2], the arcsine principal interval.
  • [0.5 marks] Therefore arcsin(2x√(1−x²))=2θ=2arcsin x.

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