Maths · Inverse trigonometrical functions and their properties
The set of values of ' for which the formula \) is true, is
The set of values of ' \( x^{\prime} \) for which the formula \( 2 \sin ^{-1} x=\sin ^{-1}(2 x \sqrt{1-x^{2}}) \) is true, is
- A. (-1,0)
- B. [0,1]
- C. \( \left[-\frac{\sqrt{3}}{2}, \frac{\sqrt{3}}{2}\right] \)
- D. \( \left[-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right] \)
Step-by-step solution
Let θ = sin⁻¹ x ⇒ x = sin θ. Then RHS = sin⁻¹(2 sin θ cos θ) = sin⁻¹(sin 2θ). The equation becomes 2θ = sin⁻¹(sin 2θ). This holds if and only if 2θ ∈ [-π/2, π/2], i.e., θ ∈ [-π/4, π/4]. Since θ = sin⁻¹ x has range [-π/2, π/2], the condition reduces to x = sin θ ∈ [-1/√2, 1/√2].
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