Question

If \(\cos\left(2\sin^{-1}\alpha\right) = \frac{47}{72}\), where \(0 < \alpha < 1\), then the value of \(\alpha\) is

• A. \(\frac{5}{12}\)
• B. \(\frac{7}{12}\)
• C. \(\frac{12}{13}\)
• D. \(\frac{5}{13}\)
• E. \(\frac{7}{13}\)

Hint:

\(\cos(2\sin^{-1} x) = 1 - 2x^2\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Let \(\theta = \sin^{-1} \alpha \Rightarrow \sin \theta = \alpha\). Then \(\cos(2\sin^{-1} \alpha) = \cos 2\theta = 1 - 2\sin^2 \theta = 1 - 2\alpha^2\).

Step 2: Detailed Explanation:
\(1 - 2\alpha^2 = \frac{47}{72} \Rightarrow 2\alpha^2 = 1 - \frac{47}{72} = \frac{72 - 47}{72} = \frac{25}{72}\)
\(\alpha^2 = \frac{25}{144} \Rightarrow \alpha = \frac{5}{12}\) (since \(\alpha$>$0\))

Step 3: Final Answer:
\(\alpha = \frac{5}{12}\).