Equation `[cot^(-1) x] + 2 [tan^(-1) x] = 0`, where `[.]` denotes the greatest integer function, is satisfied by

Options

A. `(0, cot 1)`
B. `(0, tan 1)`
C. `(tan 1, oo)`
D. `(cot 1, tan 1)`

Explanation

To solve the equation \([cot^{-1} x] + 2 [tan^{-1} x] = 0\), where \([.]\) denotes the greatest integer function, we can follow these steps:

### Step 1: Understand the Equation
The equation states that the sum of the greatest integer of the cotangent inverse of \(x\) and twice the greatest integer of the tangent inverse of \(x\) equals zero.

### Step 2: Set Up the Greatest Integer Conditions
From the equation, we can deduce:
\[ [cot^{-1} x] + 2[tan^{-1} x] = 0 \]
This implies:
\[ [cot^{-1} x] = -2[tan^{-1} x] \]
Since the greatest integer function outputs integers, we can conclude that both sides must be integers.

### Step 3: Analyze the Greatest Integer Values
1. For \([cot^{-1} x]\) to be \(0\), we must have:
\[ 0 \leq cot^{-1} x < 1 \]
This implies:
\[ cot^{-1} x = \frac{\pi}{2} - tan^{-1} x \implies 0 < x < \infty \]
Thus, \(x\) must be positive.

2. For \([tan^{-1} x]\) to be \(0\), we must have:
\[ 0 \leq tan^{-1} x < 1 \]
This implies:
\[ tan^{-1} x = 0 \implies x = 0 \]
However, since \(x\) must be positive, we consider the range:
\[ 0 < x < tan(1) \]

### Step 4: Combine the Ranges
From the analysis:
- From \([cot^{-1} x] = 0\), we have \(x > 0\).
- From \([tan^{-1} x] = 0\), we have \(0 < x < tan(1)\).

### Step 5: Determine the Final Range
The combined range for \(x\) is:
\[ 0 < x < tan(1) \]
This means \(x\) must be a positive number less than \(tan(1)\).

### Step 6: Check the Options
Now, we check the options provided in the question to find which one satisfies this range.

### Conclusion
The solution indicates that the equation is satisfied by values of \(x\) in the range \(0 < x < tan(1)\).