Solve the inequality `tan^(-1)xgtcot^(-1)cot^(-1)x`.

Explanation

To solve the inequality \( \tan^{-1} x > \cot^{-1} (\cot^{-1} x) \), we can follow these steps:

### Step 1: Rewrite the inequality
We start with the given inequality:
\[ \tan^{-1} x > \cot^{-1} (\cot^{-1} x) \]

### Step 2: Use the identity for cotangent
We know that:
\[ \tan^{-1} x = \cot^{-1} \left( \frac{1}{x} \right) \]
Thus, we can rewrite the inequality as:
\[ \cot^{-1} \left( \frac{1}{x} \right) > \cot^{-1} (\cot^{-1} x) \]

### Step 3: Simplify the right side
The term \( \cot^{-1} (\cot^{-1} x) \) simplifies to \( x \) because the cotangent and its inverse cancel each other out. Therefore, we have:
\[ \cot^{-1} \left( \frac{1}{x} \right) > x \]

### Step 4: Analyze the cotangent function
Since \( \cot^{-1} x \) is a decreasing function, we can reverse the inequality when we apply it. Thus, we can write:
\[ \frac{1}{x} < \cot^{-1} x \]

### Step 5: Rearranging the inequality
This gives us:
\[ \cot^{-1} x > \frac{1}{x} \]

### Step 6: Graphical interpretation
To understand the behavior of the functions \( \cot^{-1} x \) and \( \frac{1}{x} \), we can analyze their graphs. The graph of \( \cot^{-1} x \) approaches \( 0 \) as \( x \) approaches \( \infty \) and approaches \( \frac{\pi}{2} \) as \( x \) approaches \( 0 \). The graph of \( \frac{1}{x} \) is a hyperbola that approaches \( 0 \) as \( x \) approaches \( \infty \) and approaches \( \infty \) as \( x \) approaches \( 0 \).

### Step 7: Finding the solution
From the graphical analysis, we can see that \( \cot^{-1} x \) will always be less than \( \frac{1}{x} \) for positive values of \( x \). Therefore, there are no values of \( x \) for which \( \cot^{-1} x > \frac{1}{x} \).

### Conclusion
Thus, the inequality \( \tan^{-1} x > \cot^{-1} (\cot^{-1} x) \) has no solution.

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