Question

The value of \( \sin \left( 2 \cos^{-1} \left( \frac{5}{12} \right) + \sin^{-1} \left( \frac{5}{12} \right) \right) \) is equal to

• A. \( \frac{5}{12} \)
• B. \( \frac{12}{13} \)
• C. \( \frac{5}{13} \)
• D. \( \frac{10}{13} \)
• E. \( \frac{5}{6} \)

Hint:

Use trigonometric identities for sum and double-angle formulas to simplify complex expressions.

Answer 1

The Correct Option is A

Step 1: Define the Terms Let: \[ \theta = \cos^{-1} \left( \frac{5}{12} \right) \] Using the identity: \[ \sin^2 \theta + \cos^2 \theta = 1 \] \[ \sin^2 \theta = 1 - \left( \frac{25}{144} \right) \] \[ \sin^2 \theta = \frac{119}{144} \Rightarrow \sin \theta = \frac{\sqrt{119}}{12} \] Step 2: Compute \( 2 \cos^{-1} \left( \frac{5}{12} \right) \) Using the double angle identity: \[ \cos 2\theta = 2\cos^2 \theta - 1 \] \[ \cos 2\theta = 2 \left( \frac{25}{144} \right) - 1 = \frac{50}{144} - 1 = \frac{-94}{144} \] Step 3: Compute \( \sin(2\theta + \sin^{-1} (\frac{5}{12})) \) Using the identity: \[ \sin(A + B) = \sin A \cos B + \cos A \sin B \] \[ \sin \left( 2\theta + \sin^{-1} \left( \frac{5}{12} \right) \right) = \sin 2\theta \cos \sin^{-1} \frac{5}{12} + \cos 2\theta \sin \sin^{-1} \frac{5}{12} \] After simplification, \[ \sin \left( 2\theta + \sin^{-1} \left( \frac{5}{12} \right) \right) = \frac{5}{12} \] 
Final Answer: \[ \boxed{\frac{5}{12}} \]