Step 1: Understanding the Concept:
This problem requires the step-by-step evaluation of nested trigonometric and inverse trigonometric functions.
The evaluation proceeds from the innermost parentheses outward.
The most critical part is correctly identifying the principal values for the inverse functions:
- \(\sin^{-1}(x)\) results in an angle \(\theta \in [-\pi/2, \pi/2]\).
- \(\cot^{-1}(x)\) results in an angle \(\theta \in (0, \pi)\).
Substituting these standard values correctly at each step ensures the final numerical result is accurate.
Step 2: Key Formula or Approach:
The steps are as follows:
1. Evaluate the standard angle for \(\sin^{-1}(1/2)\).
2. Multiply that angle by 2.
3. Compute the cosine of the resulting angle.
4. Multiply the cosine value by 2 to get a real number.
5. Determine the inverse cotangent of that real number.
Step 3: Detailed Explanation:
Let's perform the calculation carefully:
Inner Term 1: \(\sin^{-1}(1/2)\).
We know that \(\sin(\pi/6) = 1/2\). Since \(\pi/6\) is in the principal range \([-\pi/2, \pi/2]\), we have:
\[ \sin^{-1}(1/2) = \frac{\pi}{6} \]
Inner Term 2: \(2 \times \sin^{-1}(1/2)\).
\[ 2 \times \frac{\pi}{6} = \frac{\pi}{3} \]
Inner Term 3: \(\cos(\pi/3)\).
The cosine of \(60^\circ\) or \(\pi/3\) is a standard value:
\[ \cos \frac{\pi}{3} = \frac{1}{2} \]
Inner Term 4: \(2 \cos(\pi/3)\).
\[ 2 \times \frac{1}{2} = 1 \]
Outer Term: \(\cot^{-1}(1)\).
We need to find an angle \(\theta \in (0, \pi)\) such that \(\cot \theta = 1\).
We know that \(\cot(\pi/4) = 1\).
Therefore, \(\cot^{-1}(1) = \frac{\pi}{4}\).
This angle \(\pi/4\) is the final value of the entire expression.
Step 4: Final Answer:
The value of the expression is \(\pi/4\).
This matches Option (B).