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

Concept:
If \[ \csc^{-1}(\sqrt{2})=\theta \] then \[ \csc\theta=\sqrt{2} \] which means \[ \sin\theta=\frac{1}{\sqrt{2}} \] ip

Step 1: Find the angle \(\csc^{-1}(\sqrt{2})\).
Since \[ \sin\theta=\frac{1}{\sqrt{2}} \] we get \[ \theta=\frac{\pi}{4} \] So, \[ \csc^{-1}(\sqrt{2})=\frac{\pi}{4} \] ip

Step 2: Substitute inside the cosine.
\[ 2\cos\left(2\csc^{-1}(\sqrt{2})\right) = 2\cos\left(2\cdot \frac{\pi}{4}\right) = 2\cos\left(\frac{\pi}{2}\right) = 2\cdot 0 = 0 \] ip

Step 3: Evaluate the inverse cotangent.
\[ \cot^{-1}(0)=\frac{\pi}{2} \] ip Hence, the correct answer is:
\[ \boxed{(A)\ \frac{\pi}{2}} \]