NCERT Class 12 · Ch 2 Exercise 2.2 · Q1
NCERT Class 12 Maths · Chapter 2 Inverse Trigonometric Functions · Exercise 2.2 Question 1
Prove 3sin⁻¹x=sin⁻¹(3x−4x³) for −1/2≤x≤1/2.
💡 Concept Hints
- Recall the principal range of the inverse function before simplifying.
- Find an angle with the required trigonometric value, then verify it belongs to the principal range.
✅ Step-by-Step Solution
Final Answer: The stated identity holds on the given interval.
- Put θ=sin⁻¹x. Since −1/2≤x≤1/2, θ∈[−π/6,π/6].
- The triple-angle identity gives sin 3θ=3sin θ−4sin³θ=3x−4x³.
- Here 3θ∈[−π/2,π/2], so applying the principal sine inverse gives sin⁻¹(3x−4x³)=3θ=3sin⁻¹x.
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