To solve the given assertion and reason, we will analyze both statements step by step.
**Step 1: Analyze Assertion (A)**
The assertion states that the range of \( \cot^{-1} x \) is \( (0, \pi) \).
- The cotangent function, \( \cot x \), is defined for \( x \) in the interval \( (0, \pi) \) and takes all real values.
- Therefore, the inverse function \( \cot^{-1} x \) will have a range that corresponds to the domain of \( \cot x \), which is \( (0, \pi) \).
**Conclusion for Step 1:**
Thus, the assertion (A) is **true**.
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**Step 2: Analyze Reason (R)**
The reason states that the domain of \( \tan^{-1} x \) is \( \mathbb{R} \).
- The tangent function, \( \tan x \), has a range of all real numbers \( \mathbb{R} \) as \( x \) varies over its domain, which excludes odd multiples of \( \frac{\pi}{2} \).
- Consequently, the inverse function \( \tan^{-1} x \) is defined for all real numbers, confirming that its domain is indeed \( \mathbb{R} \).
**Conclusion for Step 2:**
Thus, the reason (R) is also **true**.
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**Step 3: Determine the relationship between A and R**
Now we need to check if the reason (R) is the correct explanation for the assertion (A).
- The assertion (A) discusses the range of \( \cot^{-1} x \), while the reason (R) discusses the domain of \( \tan^{-1} x \).
- There is no direct relationship between the range of \( \cot^{-1} x \) and the domain of \( \tan^{-1} x \); they are independent properties of different functions.
**Conclusion for Step 3:**
While both statements are true, the reason (R) does not provide a correct explanation for the assertion (A).
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**Final Conclusion:**
Both assertion (A) and reason (R) are true, but (R) is not the correct explanation for (A). Therefore, the answer is that both A and R are true, but R is not the correct explanation of A.
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