Question

\(\cos^{-1}\{\cos 2\cot^{-1}(\sqrt{2} - 1)\}\) is equal to

• A. \(\sqrt{2} - 1\)
• B. \(\frac{\pi}{4}\)
• C. \(\frac{3\pi}{4}\)
• D. 0

Hint:

\(\cot^{-1}(\sqrt{2} - 1) = \frac{3\pi}{8}\) and \(\cos^{-1}(\cos \theta) = \theta\) for \(\theta \in [0, \pi]\).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Simplify \(\cot^{-1}(\sqrt{2} - 1)\) to a known angle.
Step 2: Detailed Explanation:
\(\sqrt{2} - 1 = \tan 22.5^\circ = \tan \frac{\pi}{8}\)
So \(\cot^{-1}(\sqrt{2} - 1) = \frac{\pi}{2} - \frac{\pi}{8} = \frac{3\pi}{8}\)
Then \(2\cot^{-1}(\sqrt{2} - 1) = \frac{3\pi}{4}\)
\(\cos^{-1}\{\cos \frac{3\pi}{4}\} = \frac{3\pi}{4}\) (since \(\frac{3\pi}{4} \in [0, \pi]\))
Step 3: Final Answer:
\(\frac{3\pi}{4}\).