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

Step 1: Understanding the Concept:
Solve the expression from the innermost bracket outward. Evaluate the inverse sine first, then the cosine, then the inverse tangent.
Step 2: Formula Application:
1. $\sin^{-1}(1/2) = \pi/6$
2. $2 \times (\pi/6) = \pi/3$
3. $\cos(\pi/3) = 1/2$
Step 3: Explanation:
The expression simplifies to: $$\tan^{-1} [ 2 \times (1/2) ] = \tan^{-1} [1]$$ Since $\tan(\pi/4) = 1$ and $\pi/4$ is in the principal range: $\tan^{-1}(1) = \pi/4$.
Step 4: Final Answer:
The value is $\pi/4$.