Question

\( \tan^{-1} \left[ 2\cos \left( 2\sin^{-1} \frac{1}{2} \right) \right] = \)_____

• A. \( \frac{\pi}{4} \)
• B. \( \frac{3\pi}{4} \)
• C. \( -\frac{\pi}{4} \)
• D. \( -\frac{3\pi}{4} \)

Hint:

Always evaluate inner inverse trigonometric function first.

Answer 1

The Correct Option is A

Concept: Use identity: \[ \cos(2\theta) = 1 - 2\sin^2 \theta \]
Step 1: Find inner value. \[ \sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6} \]
Step 2: \[ 2\sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{3} \]
Step 3: \[ \cos \frac{\pi}{3} = \frac{1}{2} \] \[ 2 \times \frac{1}{2} = 1 \]
Step 4: \[ \tan^{-1}(1) = \frac{\pi}{4} \]