Question

\(\cot^{-1} (2 \cos(2 \text{cosec}^{-1}(\sqrt{2}))) = \dots\)

• A. \(\frac{\pi}{2}\)
• B. \(\frac{\pi}{3}\)
• C. \(\frac{\pi}{4}\)
• D. \(0\)

Hint:

In inverse trigonometric questions, always convert the inverse function into an angle first, then simplify step by step.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
We evaluate the expression starting from the innermost inverse trigonometric function.
Step 2: Key Formula or Approach:
Basic trigonometric values: \(\text{cosec}(\frac{\pi}{4}) = \sqrt{2}\) and \(\cos(\frac{\pi}{2}) = 0\).
Step 3: Detailed Explanation:
Let \(\theta = \text{cosec}^{-1}(\sqrt{2})\). Since \(\text{cosec} \frac{\pi}{4} = \sqrt{2}\), \(\theta = \frac{\pi}{4}\).
The expression becomes:
\[ \cot^{-1} (2 \cos(2 \cdot \frac{\pi}{4})) \] Simplify the argument inside the cosine function:
\[ = \cot^{-1} (2 \cos(\frac{\pi}{2})) \] Using \(\cos \frac{\pi}{2} = 0\):
\[ = \cot^{-1} (2 \cdot 0) = \cot^{-1}(0) \] Since \(\cot \frac{\pi}{2} = 0\), it follows that \(\cot^{-1}(0) = \frac{\pi}{2}\).
Step 4: Final Answer:
The result is \(\frac{\pi}{2}\).