Question:
\( \cos^{-1}\{\cos 2\cot^{-1}(\sqrt{2}-1)\} \) is equal to
\( \cos^{-1}\{\cos 2\cot^{-1}(\sqrt{2}-1)\} \) is equal to
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Always check principal value range in inverse trigonometric functions.
Updated On: Apr 23, 2026
- $\sqrt{2}-1$
- $\frac{\pi}{4}$
- $\frac{3\pi}{4}$
- $0$
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The Correct Option is C
Solution and Explanation
Concept:
Principal value of $\cos^{-1}(\cos \theta)$ lies in $[0,\pi]$.
Step 1: Let $\theta = \cot^{-1}(\sqrt{2}-1)$.
Step 2: Convert to tangent form.
\[ \tan \theta = \frac{1}{\sqrt{2}-1} = \sqrt{2}+1 \]
Step 3: Recognize standard value.
\[ \tan \theta = \sqrt{2}+1 \Rightarrow \theta = \frac{3\pi}{8} \]
Step 4: Compute expression.
\[ 2\theta = \frac{3\pi}{4} \]
Step 5: Apply principal value.
\[ \cos^{-1}(\cos 2\theta) = \frac{3\pi}{4} \] Conclusion:
Answer = $\frac{3\pi}{4}$
Step 1: Let $\theta = \cot^{-1}(\sqrt{2}-1)$.
Step 2: Convert to tangent form.
\[ \tan \theta = \frac{1}{\sqrt{2}-1} = \sqrt{2}+1 \]
Step 3: Recognize standard value.
\[ \tan \theta = \sqrt{2}+1 \Rightarrow \theta = \frac{3\pi}{8} \]
Step 4: Compute expression.
\[ 2\theta = \frac{3\pi}{4} \]
Step 5: Apply principal value.
\[ \cos^{-1}(\cos 2\theta) = \frac{3\pi}{4} \] Conclusion:
Answer = $\frac{3\pi}{4}$
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