Class 12 Inverse Trigonometric Functions · Short Answer2 Marksinverse_trigonometric_identities
Class 12 Maths · Inverse Trigonometric Functions · Short Answer #61
Prove \(3\sin^{-1}x=\sin^{-1}(3x-4x^3)\) for \(-1/2\le x\le1/2\).
✅ Step-by-Step Solution
Final Answer: Identity proved on [−1/2,1/2].
- Put θ=arcsin x; then θ∈[−π/6,π/6].
- The triple-angle identity gives 3x−4x³=sin3θ and 3θ∈[−π/2,π/2].
- Hence arcsin(3x−4x³)=3θ=3arcsin x.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Put θ=arcsin x; then θ∈[−π/6,π/6].
- [1 mark] The triple-angle identity gives 3x−4x³=sin3θ and 3θ∈[−π/2,π/2].
- [0.5 marks] Hence arcsin(3x−4x³)=3θ=3arcsin x.
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