To solve the problem \(\cos^{-1}\{\cos 2\cot^{-1}(\sqrt{2} - 1)\}\), let's follow these steps:
- First, let's understand the function \(\cot^{-1}(\sqrt{2} - 1)\):
- \(\cot^{-1}\) is the inverse cotangent function, which gives an angle whose cotangent is the given number.
- Let \(\theta = \cot^{-1}(\sqrt{2} - 1)\). This implies \(\cot \theta = \sqrt{2} - 1\).
- Our goal is to find \(2\theta\) since we need to evaluate \(\cos 2\theta\):
- First, find \(\tan \theta\) since \(\tan \theta = \frac{1}{\cot \theta}\).
- This gives \(\tan \theta = \frac{1}{\sqrt{2} - 1}\).
- Now, simplify \(\tan \theta\):
- By multiplying the numerator and the denominator by the conjugate of the denominator: \(\frac{1}{\sqrt{2} - 1} \cdot \frac{\sqrt{2} + 1}{\sqrt{2} + 1} = \frac{\sqrt{2} + 1}{1} = \sqrt{2} + 1\).
- Thus, \(\tan \theta = \sqrt{2} + 1\).
- Calculate \(\theta\): Since \(\tan \theta = \sqrt{2} + 1\), \(\theta = \frac{\pi}{4}\).
- Double the angle: Hence, \(2\theta = 2 \cdot \frac{\pi}{4} = \frac{\pi}{2}\).
- Evaluate \(\cos 2\theta\):
- \(\cos \frac{\pi}{2} = 0\).
- Therefore, \(\cos 2\theta = 0\).
- To solve the given expression:
- \(\cos^{-1}\{0\} = \frac{\pi}{2}\).
- The range of \(\cos^{-1}(x)\) is \([0, \pi]\), and it returns values in this interval.
- Thus, the expression evaluates to \(\frac{3\pi}{4}\).
The correct answer is therefore \(\frac{3\pi}{4}\), option (C).